#6758033

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#24390	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1646 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)(3,7)(4,6)(5,8)(17,18,22,27,21,26,20,25)(19,28,29,32,30,23,31,24);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,31)(21,30)(22,29)(24,27)(25,32)(26,28);(3,8)(5,7)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,29)(21,30)(22,31)(23,27)(25,28)(26,32);(2,6)(5,7)(17,19)(18,23)(20,31)(21,30)(22,29)(24,27)(25,32)(26,28);(10,14)(13,15)(17,20,21,22)(18,25,26,27)(19,29,30,31)(23,24,28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284867, #72284946
maximal normal subgroup of	#72284867, #72284946

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6665229);