#6718195

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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1589 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)(17,18,22,24)(19,29,31,25)(20,26,23,28)(21,27,30,32);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(18,24)(19,30)(20,23)(21,31)(25,32)(27,29);(10,15)(14,16)(17,19,22,31)(18,25,24,29)(20,30,23,21)(26,27,28,32);(2,6)(5,7)(17,20,22,23)(18,26,24,28)(19,30,31,21)(25,27,29,32);(3,8)(5,7)(9,12,13,11)(10,16,15,14)(17,21)(18,27)(19,20)(22,30)(23,31)(24,32)(25,28)(26,29)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=i,e^2=gi,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, 6625391);