#6473460

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1470 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

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, 6380656);