#6477112

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469956
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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1668 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:

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

Polycyclic presentation:

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

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, 6384308);