#6548432

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1540 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:

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

Polycyclic presentation:

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

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, 6455628);