#6555432

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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1530 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,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, 6462628);