#7171676

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,5)(6,7)(10,13)(11,15)(17,18,23,25,21,26,22,27)(19,28,30,31,20,24,29,32);(1,2,4,5)(3,7,8,6)(10,13)(11,15)(17,19)(18,24)(20,21)(22,29)(23,30)(25,32)(26,28)(27,31);(9,10)(11,14)(12,13)(15,16)(18,25)(19,20)(22,23)(24,31)(26,27)(28,32);(1,3,4,8)(2,6,5,7)(10,13)(14,16)(17,20,21,19)(18,24,26,28)(22,30,23,29)(25,32,27,31);(9,11,12,15)(10,14,13,16)(18,26)(19,20)(25,27)(29,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,d^2=f,e^2=hi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7078872);