#6461891

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1832 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)(3,8)(4,6)(5,7)(17,18,23,26,22,28,21,27)(19,29,31,24,33,42,32,38)(20,25,36,40,34,39,35,41)(30,43,45,37,44,48,46,47);(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,30)(21,31)(22,33)(23,32)(25,37)(26,29)(27,42)(28,38)(34,44)(35,45)(36,46)(39,47)(40,43)(41,48);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,30)(21,35)(22,34)(23,36)(24,37)(26,40)(27,41)(28,39)(29,43)(31,45)(32,46)(33,44)(38,47)(42,48);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(20,34)(25,39)(30,44)(35,36)(37,47)(40,41)(43,48)(45,46);(17,21,22,23)(18,26,28,27)(19,31,33,32)(20,35,34,36)(24,29,38,42)(25,40,39,41)(30,45,44,46)(37,43,47,48)

Polycyclic presentation:

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

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, 6369087);