#6536114

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

1865 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)(3,8)(4,6)(5,7)(17,18,22,27)(19,28,30,24)(20,33,35,40)(21,26,23,36)(25,37,41,34)(29,44,45,48)(31,43,32,39)(38,46,42,47);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,27,28)(20,29,35,45)(21,31,23,32)(25,38,41,42)(26,39,36,43)(33,48,40,44)(34,46,37,47);(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,37)(24,38)(26,40)(27,41)(28,42)(30,45)(31,46)(32,47)(33,36)(39,44)(43,48);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,30)(24,28)(29,45)(31,32)(38,42)(39,43)(44,48)(46,47);(11,15)(14,16)(17,21,22,23)(18,26,27,36)(19,31,30,32)(20,34,35,37)(24,39,28,43)(25,40,41,33)(29,46,45,47)(38,44,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,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, 6443310);