#6491968

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1553 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^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=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6399164);