#6549428

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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1564 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,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, 6456624);