#6874844

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1437 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)(17,18,22,27,21,26,35,24)(19,28,31,37,30,23,36,39)(20,32,34,40,33,25,41,42)(29,38,45,48,44,43,46,47);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,14)(17,19,21,30)(18,23,26,28)(20,29,33,44)(22,31,35,36)(24,37,27,39)(25,38,32,43)(34,45,41,46)(40,47,42,48);(3,8)(5,7)(10,14)(15,16)(18,24)(22,35)(23,37)(25,40)(26,27)(28,39)(31,36)(32,42)(34,41)(38,47)(43,48)(45,46);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,38)(24,40)(26,32)(27,42)(28,43)(30,44)(31,45)(35,41)(36,46)(37,47)(39,48);(9,12,13,11)(10,15,14,16)(18,26)(19,30)(20,33)(24,27)(31,36)(34,41)(38,43)(47,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,d^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6782040);