#6553238

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1420 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)(17,18,22,27,23,26,21,34)(19,24,29,40,30,38,35,39)(20,31,32,41,33,42,36,25)(28,43,45,47,46,48,44,37);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,28)(21,35)(22,29)(23,30)(25,37)(26,38)(27,40)(31,43)(32,45)(33,46)(34,39)(36,44)(41,47)(42,48);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,23,33)(18,25,26,41)(19,28,30,46)(21,32,22,36)(24,37,38,47)(27,42,34,31)(29,44,35,45)(39,43,40,48);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(18,26)(24,38)(25,41)(27,34)(31,42)(37,47)(39,40)(43,48);(11,15)(14,16)(17,21,23,22)(18,27,26,34)(19,29,30,35)(20,32,33,36)(24,39,38,40)(25,42,41,31)(28,44,46,45)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,d^2=f,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, 6460434);