#6549444

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

1566 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)(17,18,22,26,23,28,21,27)(19,24,30,39,32,40,31,38)(20,33,34,42,35,41,36,25)(29,43,45,47,46,48,44,37);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,32)(18,24,28,40)(20,29,35,46)(21,31,22,30)(25,37,42,47)(26,39,27,38)(33,43,41,48)(34,45,36,44);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,37)(26,41)(27,33)(28,42)(30,44)(31,45)(32,46)(38,43)(39,48)(40,47);(9,11)(10,14)(12,15)(13,16)(17,21,23,22)(18,26,28,27)(19,30,32,31)(20,34,35,36)(24,38,40,39)(25,41,42,33)(29,44,46,45)(37,43,47,48);(11,15)(14,16)(17,21,23,22)(18,27,28,26)(19,31,32,30)(20,34,35,36)(24,38,40,39)(25,33,42,41)(29,45,46,44)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6456640);