#6910162

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907945
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	#37681	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1934 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)(19,22)(20,28)(21,25)(23,27)(24,30)(26,31)(29,32);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,18)(20,26,29,24)(21,25,23,27)(28,30,32,31);(3,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,20,23,30,22,29,21,31)(18,24,25,28,19,26,27,32);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(18,19)(20,29)(25,27)(30,31);(2,5)(6,8)(10,14)(13,15)(17,21,22,23)(18,25,19,27)(20,30,29,31)(24,32,26,28)

Polycyclic presentation:

2^9|b^2=f,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,d^b=di,e^b=efh,d^c=dfh,e^c=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, 6817358);