#72300524

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 9749 conjugacy classes of subgroups; only the 3301 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|b^2=gi,d^2=h,e^2=gi,b^a=bg,c^a=cgi,c^b=cgi,d^a=dh,d^b=dh,d^c=dghij,e^a=egi,e^c=egi,e^d=eghi,f^a=fh,f^c=fgj,f^d=fh

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300575
maximal normal subgroup of	#72300575

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