#6819548

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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1375 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)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,30,29)(20,24,21,25)(27,32,28,31);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,31)(21,32)(22,30)(24,27)(25,28)(26,29);(1,4,5,3)(2,6,7,8)(10,14)(11,16)(17,20,26,25,22,21,18,24)(19,27,23,32,30,28,29,31);(10,14)(13,15)(17,18,22,26)(19,23,30,29)(20,24,21,25)(27,32,28,31);(17,21,26,24,22,20,18,25)(19,28,23,31,30,27,29,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=ghi,d^2=i,e^2=fhi,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, 6726744);