#6883837

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	#38024	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1613 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)(17,18,21,23)(19,26,25,22)(20,28,29,24)(27,30,31,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(18,23)(19,25)(24,28)(27,31);(2,6)(3,8)(10,14)(15,16)(17,19,22,18,21,25,26,23)(20,27,30,24,29,31,32,28);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,28)(25,31)(26,32);(9,12,13,11)(10,15,14,16)(20,29)(24,28)(27,31)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6791033);