#6550068

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1464 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)(3,7)(4,8)(5,6)(17,18,23,27,22,26,21,28)(19,29,32,39,31,24,33,41)(20,34,35,25,36,42,37,40)(30,44,47,43,45,48,46,38);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,24,26,29)(20,30,36,45)(21,33,23,32)(25,38,40,43)(27,41,28,39)(34,48,42,44)(35,47,37,46);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,36)(18,25,26,40)(19,30,31,45)(21,35,23,37)(24,38,29,43)(27,34,28,42)(32,46,33,47)(39,44,41,48);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(18,26)(19,31)(25,40)(27,28)(30,45)(32,33)(34,42)(46,47);(11,15)(14,16)(17,21,22,23)(18,27,26,28)(19,32,31,33)(20,35,36,37)(24,39,29,41)(25,34,40,42)(30,46,45,47)(38,44,43,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,e^b=ei,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, 6457264);