#6868341

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#31057	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1591 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)(17,18,22,27,23,26,19,24)(20,29,28,21,30,31,32,25);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,22)(18,24,26,27)(20,28,30,32)(21,31,25,29);(10,14)(15,16)(17,20,23,30)(18,25,26,21)(19,28,22,32)(24,29,27,31);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21)(18,20)(19,29)(22,31)(23,25)(24,32)(26,30)(27,28);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(18,26)(21,25)(24,27)(29,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6775537);