#6637124

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636908
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	#29732	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1528 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,23,27,31)(20,29,25,21)(26,28,30,32);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,27)(18,23,24,31)(20,30,25,26)(21,28,29,32);(11,15)(13,16)(17,20,21,18,22,25,29,24)(19,26,28,31,27,30,32,23);(2,6)(3,8)(11,15)(13,16)(19,27)(23,31)(26,30)(28,32);(2,6)(5,7)(10,14)(13,16)(18,24)(19,27)(20,25)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6544320);