#7006206

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2323 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308532, #72308909
maximal normal subgroup of	#72308532, #72308909

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6913402);