#6718200

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,h^e=hi

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, 6625396);