#6909652

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#23850	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1885 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,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=efhi,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6816848);