#6495789

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469983
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	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 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)(3,8)(4,6)(5,7)(11,15)(14,16)(17,18,22,27,23,26,21,25)(19,28,30,20,31,32,29,24);(9,10)(11,14)(12,13)(15,16)(17,19,23,31)(18,24,26,20)(21,30,22,29)(25,28,27,32);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(17,20,23,24)(18,19,26,31)(21,28,22,32)(25,29,27,30);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,21,23,22)(18,25,26,27)(19,29,31,30)(20,28,24,32);(9,11,12,15)(10,14,13,16)(18,26)(20,24)(25,27)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6402985);