#6763801

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37680	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1673 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18)(19,20)(21,25)(22,26)(23,32)(24,27)(28,30)(29,31);(1,3)(2,5)(4,7)(6,8)(9,10,12,13)(11,14,15,16)(17,19)(18,20)(21,22)(23,28)(24,29)(25,26)(27,31)(30,32);(2,6)(3,7)(10,13)(14,16)(17,20,23,25,24,31,22,30)(18,21,26,19,27,28,32,29);(9,11,12,15)(10,14,13,16)(17,21,24,28)(18,25,27,30)(19,22,29,23)(20,26,31,32);(3,7)(5,8)(17,22,24,23)(18,26,27,32)(19,28,29,21)(20,30,31,25)

Polycyclic presentation:

2^9|b^2=f,c^2=fghi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6670997);