#6536554

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	#23390	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1804 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)(3,6)(4,5)(7,8)(17,18)(19,28)(20,33)(21,26)(22,38)(23,27)(24,31)(25,36)(29,46)(30,40)(32,45)(34,44)(35,43)(37,42)(39,48)(41,47);(9,10)(11,14)(12,13)(15,16)(17,19,22,31,23,32,21,30)(18,24,26,28,27,40,38,45)(20,34,35,48,36,29,37,47)(25,41,43,46,33,39,42,44);(2,5)(6,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,37)(22,35)(23,36)(24,39)(26,43)(27,33)(28,44)(30,48)(31,47)(32,34)(38,42)(40,41)(45,46);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16);(11,15)(14,16)(17,21,23,22)(18,26,27,38)(19,30,32,31)(20,35,36,37)(24,28,40,45)(25,42,33,43)(29,47,34,48)(39,46,41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6443750);