#6441073

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1759 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)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(19,20)(21,27)(23,24)(26,30)(28,29)(31,32);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,30)(25,31)(26,32);(17,20,22,29)(18,24,25,32)(19,27,28,21)(23,30,31,26);(18,25)(23,31)(24,32)(26,30)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

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, 6348269);