#6432693

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339889);