#6454994

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1504 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)(11,14)(12,13)(15,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,43)(28,37)(29,38)(30,36)(31,41)(32,39)(33,42)(34,40)(35,44)(45,48)(46,47);(1,3,4,8)(2,5,6,7)(17,19)(18,23)(20,31)(21,29)(22,27)(24,38)(25,41)(26,43)(28,33)(30,34)(32,46)(35,45)(36,40)(37,42)(39,47)(44,48);(10,13)(14,16)(18,24)(19,27)(22,34)(23,36)(25,39)(28,31)(29,30)(33,35)(37,47)(38,43)(41,48)(45,46);(17,20,21,32)(18,25,24,39)(19,28,29,45)(22,35,34,33)(23,37,38,48)(26,44,40,42)(27,31,30,46)(36,47,43,41);(9,11,12,15)(10,14,13,16)(18,24)(23,38)(25,39)(26,40)(36,43)(37,48)(41,47)(42,44)

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi

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, 6362190);