#6549416

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fi,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, 6456612);