#6634780

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632008
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	#35005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1692 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)(9,10,14,15,13,11,12,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(1,3)(2,5)(4,7)(6,8)(10,11)(15,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31);(3,7)(5,8)(9,11,13,10)(12,16,14,15)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31);(3,7)(5,8)(9,12,13,14)(10,15,11,16)(18,24)(19,27)(23,30)(26,32);(2,6)(5,8)(18,24)(22,25)(23,30)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6541976);