#7156258

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7063454);