#6681815

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#36019	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1493 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)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,25)(19,27,30,23)(20,32,33,24)(22,26,35,40)(28,38,31,36)(29,37,44,41)(34,45,46,39)(42,47,43,48);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(17,19,21,30)(18,23,25,27)(20,29,33,44)(22,28,35,31)(24,37,32,41)(26,36,40,38)(34,42,46,43)(39,47,45,48);(10,14)(13,16)(19,28)(22,35)(23,36)(26,40)(27,38)(29,42)(30,31)(34,46)(37,47)(39,45)(41,48)(43,44);(1,4,6,5)(2,7,3,8)(17,20)(18,24)(19,29)(21,33)(22,34)(23,37)(25,32)(26,39)(27,41)(28,42)(30,44)(31,43)(35,46)(36,47)(38,48)(40,45);(1,5,6,4)(2,7,3,8)(19,28)(20,33)(22,35)(23,36)(24,32)(26,40)(27,38)(29,43)(30,31)(37,48)(41,47)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6589011);