#6449304

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1688 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=g,e^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, 6356500);