#6635595

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

1755 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,22,25,21,27,20,26)(19,23,28,32,30,24,29,31);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,32)(26,31);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,18,21,27)(19,24,30,23)(20,25,22,26)(28,32,29,31);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,31)(26,32);(2,6)(5,8)(17,20,21,22)(18,25,27,26)(19,28,30,29)(23,31,24,32)

Polycyclic presentation:

2^9|a^2=fgi,c^2=fi,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, 6542791);