#6492968

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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1460 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6400164);