#6468002

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=h,d^2=h,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, 6375198);