#6553132

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 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)(17,18,21,27,22,26,37,25)(19,28,32,41,30,23,31,39)(20,33,36,40,35,42,34,24)(29,44,46,38,45,48,47,43);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,29)(21,32)(22,30)(24,38)(25,41)(26,28)(27,39)(31,37)(33,48)(34,47)(35,45)(36,46)(40,43)(42,44);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,22,35)(18,24,26,40)(19,29,30,45)(21,34,37,36)(23,38,28,43)(25,33,27,42)(31,46,32,47)(39,44,41,48);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(19,30)(23,28)(29,45)(31,32)(38,43)(39,41)(44,48)(46,47);(11,15)(14,16)(17,21,22,37)(18,25,26,27)(19,31,30,32)(20,34,35,36)(23,39,28,41)(24,33,40,42)(29,46,45,47)(38,44,43,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,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, 6460328);