#6446573

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1399 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,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, 6353769);