#6642923

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632082
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	#37300	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1743 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6550119);