#6549202

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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1385 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,36,22,26,21,27)(19,24,29,39,31,38,30,40)(20,32,33,25,35,42,34,41)(28,43,46,37,45,48,44,47);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,22,31)(18,24,26,38)(20,28,35,45)(21,30,23,29)(25,37,41,47)(27,40,36,39)(32,43,42,48)(33,46,34,44);(10,11)(14,15)(17,20)(18,25)(19,28)(21,33)(22,35)(23,34)(24,37)(26,41)(27,42)(29,44)(30,46)(31,45)(32,36)(38,47)(39,43)(40,48);(1,4,5,3)(2,6,7,8)(12,16)(14,15)(18,26)(24,38)(25,41)(27,36)(32,42)(37,47)(39,40)(43,48);(9,12,13,16)(10,14,11,15)(17,21,22,23)(18,27,26,36)(19,29,31,30)(20,33,35,34)(24,39,38,40)(25,42,41,32)(28,44,45,46)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6456398);