#6489985

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1373 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)(9,10)(11,14)(12,15)(13,16)(17,18,22,27)(19,29,30,24)(20,26,36,44)(21,28,23,25)(31,42,34,39)(32,40,33,38)(35,41,37,43)(45,47,46,48);(2,5)(6,8)(9,11,14,10)(12,15,13,16)(17,19)(18,24)(20,34)(21,32)(22,30)(23,33)(25,40)(26,42)(27,29)(28,38)(31,36)(35,46)(37,45)(39,44)(41,47)(43,48);(9,12,14,13)(10,15,11,16)(18,25)(19,30)(21,23)(24,38)(26,41)(27,28)(29,40)(31,34)(35,37)(39,47)(42,48)(43,44);(9,12,14,13)(10,15,11,16)(17,20,22,36)(18,26,27,44)(19,31,30,34)(21,37,23,35)(24,39,29,42)(25,41,28,43)(32,46,33,45)(38,47,40,48);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,15,11,16)(17,21,22,23)(18,25,27,28)(19,32,30,33)(20,35,36,37)(24,40,29,38)(26,43,44,41)(31,45,34,46)(39,47,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^c=ei,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, 6397181);