#6388809

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#432 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1637 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,22,27,23,26,21,35)(19,24,31,39,29,38,30,45)(20,32,33,25,34,41,36,40)(28,42,44,37,43,48,46,47);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,10)(14,15)(17,19,22,31,23,29,21,30)(18,24,27,39,26,38,35,45)(20,28,33,44,34,43,36,46)(25,37,41,48,40,47,32,42);(17,20)(18,25)(19,28)(21,36)(22,33)(23,34)(24,37)(26,40)(27,41)(29,43)(30,46)(31,44)(32,35)(38,47)(39,48)(42,45);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(18,26)(19,29)(25,40)(27,35)(28,43)(30,31)(32,41)(44,46);(1,5,6,4)(2,8,9,14)(3,11,12,13)(7,16,10,15)(19,29)(24,38)(28,43)(30,31)(37,47)(39,45)(42,48)(44,46)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=dfi,e^b=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296005);