#6468006

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1406 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)(19,24)(20,26)(21,31)(22,28)(23,27)(25,29)(30,32);(1,3,4,8)(2,5,6,7)(17,19)(18,24)(20,21)(22,29)(23,30)(25,28)(26,31)(27,32);(10,13)(14,16)(17,19,21,30,22,29,23,20)(18,25,31,26,28,24,27,32);(17,20,21,19,22,30,23,29)(18,26,31,24,28,32,27,25);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,28,31)(19,20,29,30)(24,32,25,26)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375202);