#6447500

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1669 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)(12,16)(13,15)(17,18)(19,24)(20,26)(21,27)(22,25)(23,28)(29,43)(30,38)(31,37)(32,39)(33,41)(34,42)(35,40)(36,44)(45,47)(46,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,33)(21,31)(22,32)(23,29)(25,39)(26,41)(27,37)(28,43)(30,34)(35,45)(36,46)(38,42)(40,47)(44,48);(9,12,13,11)(10,14,15,16)(18,25)(19,29)(21,23)(24,37)(26,40)(30,33)(31,32)(34,36)(38,47)(39,43)(41,48)(45,46);(17,20,22,35)(18,26,25,40)(19,30,32,46)(21,34,23,36)(24,38,39,48)(27,42,28,44)(29,33,31,45)(37,47,43,41);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,27,25,28)(19,31,32,29)(20,34,35,36)(24,37,39,43)(26,42,40,44)(30,45,46,33)(38,47,48,41)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354696);