#6546252

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1521 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,e^c=ei,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6453448);