#6817578

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806527
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1378 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)(9,10,12,14)(11,15,16,13)(17,18,22,25)(19,23,29,28)(20,24,30,21)(26,32,31,27);(1,3)(2,5)(4,8)(6,7)(17,19)(18,23)(20,27)(21,31)(22,29)(24,26)(25,28)(30,32);(9,11,12,16)(10,13,14,15)(17,20,18,24,22,30,25,21)(19,26,28,27,29,31,23,32);(2,6)(5,7)(17,18,22,25)(19,23,29,28)(20,24,30,21)(26,32,31,27);(3,8)(5,7)(17,21,25,30,22,24,18,20)(19,27,23,26,29,32,28,31)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=i,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724774);