#6548629

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1410 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6455825);