#6506208

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1383 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)(3,8)(4,6)(5,7)(17,18,22,28,23,29,21,27)(19,24,32,41,33,42,37,45)(20,34,35,44,36,26,38,25)(30,40,47,43,31,48,46,39);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,33)(18,24,29,42)(20,31,36,30)(21,37,22,32)(25,43,44,39)(26,40,34,48)(27,45,28,41)(35,46,38,47);(9,11)(10,13)(12,16)(14,15)(17,20,23,36)(18,25,29,44)(19,30,33,31)(21,35,22,38)(24,39,42,43)(26,27,34,28)(32,46,37,47)(40,41,48,45);(11,16)(13,15)(17,20)(18,26)(19,31)(21,38)(22,35)(23,36)(24,40)(25,28)(27,44)(29,34)(30,33)(32,46)(37,47)(39,45)(41,43)(42,48);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,27,29,28)(19,32,33,37)(20,35,36,38)(24,41,42,45)(25,34,44,26)(30,46,31,47)(39,48,43,40)

Polycyclic presentation:

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

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, 6413404);