#6832527

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806575
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

1371 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=fhi,d^2=i,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi

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, 6739723);