#72326953

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 9672 conjugacy classes of subgroups; only the 3285 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(2,4)(5,6)(7,10)(8,12)(9,11)(13,14)(15,17)(16,19)(18,20)(21,22)(23,26)(24,28)(25,27);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,14,18,20)(15,16,19,17)(21,23)(22,26)(24,27)(25,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,20)(18,19)(21,22)(23,24)(25,27)(26,28);(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(13,16,18,17)(14,19,20,15)(21,23,25,28)(22,26,27,24);(5,6)(7,10)(8,12)(9,11)(13,14)(15,16)(17,19)(18,20)(21,22)(23,26)(24,28)(25,27);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(13,17,18,16)(14,15,20,19)(21,24,25,26)(22,23,27,28)

Polycyclic presentation:

2^10|b^2=gh,c^2=ghj,d^2=ghi,f^2=hij,b^a=bg,c^a=ch,c^b=cgi,d^a=dh,d^b=di,d^c=dij,e^b=egh,e^c=egh,e^d=egh,f^a=fghij,f^b=fghj,f^c=fgh,f^d=fi,f^e=fhij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327062
maximal normal subgroup of	#72327062

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.7*F.8);;
gap> SetPower(coll, 3, F.7*F.8*F.10);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetPower(coll, 6, F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);