#6463504

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1571 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,27)(20,25)(21,23)(22,35)(24,30)(26,34)(28,42)(29,41)(31,46)(32,37)(33,48)(36,45)(38,44)(39,47)(40,43);(2,5)(6,8)(11,12)(14,16)(18,23)(24,36)(25,37)(26,35)(27,41)(38,39)(40,48)(42,46);(9,11,13,12)(10,14,15,16)(17,19,22,30,21,29,34,45)(18,24,26,27,23,36,35,41)(20,31,33,47,32,28,43,44)(25,39,40,46,37,38,48,42);(9,12,13,11)(10,14,15,16)(17,20,21,32)(18,25,23,37)(19,28,29,31)(22,33,34,43)(24,38,36,39)(26,40,35,48)(27,42,41,46)(30,44,45,47);(1,3,4,7)(2,6,5,8)(19,29)(20,32)(24,36)(25,37)(27,41)(30,45)(33,43)(40,48)

Polycyclic presentation:

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

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, 6370700);