#6635593

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	#23924	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1765 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)(17,18,21,25,22,26,30,31)(19,27,29,32,20,23,28,24);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,23)(20,22)(21,29)(24,25)(26,27)(28,30)(31,32);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,19,22,20)(18,24,26,32)(21,28,30,29)(23,31,27,25);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,20,22,19)(18,23,26,27)(21,28,30,29)(24,31,32,25);(2,6)(5,8)(17,21,22,30)(18,25,26,31)(19,28,20,29)(23,32,27,24)

Polycyclic presentation:

2^9|a^2=fgi,c^2=f,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, 6542789);