#7090811

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996335
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×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1671 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=ghi,d^2=f,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh

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, 6998007);