#72336588

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 10060 conjugacy classes of subgroups; only the 3309 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gj,d^2=ghi,e^2=hij,f^2=ghi,b^a=bh,c^a=cghi,c^b=cghj,d^a=dhi,d^b=dhij,d^c=dghi,e^a=eghij,e^b=eghi,e^c=eghi,e^d=egj,f^a=fj,f^c=fghi,f^d=fgij,f^e=fhj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336654
maximal normal subgroup of	#72336654

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, 1, F.7);;
gap> SetPower(coll, 2, F.7*F.10);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetPower(coll, 5, F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.7*F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);