#6448555

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

1426 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)(9,10,13,15)(11,14,12,16)(17,18)(19,24)(20,26)(21,27)(22,25)(23,28)(29,39)(30,38)(31,40)(32,37)(33,42)(34,43)(35,41)(36,44)(45,47)(46,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,22,31)(18,24,25,40)(20,33,35,45)(21,32,23,29)(26,42,41,47)(27,37,28,39)(30,34,46,36)(38,43,48,44);(9,12,13,11)(10,14,15,16)(18,25)(19,29)(21,23)(24,37)(26,41)(30,45)(31,32)(33,46)(34,36)(38,42)(39,40)(47,48);(17,20,22,35)(18,26,25,41)(19,30,31,46)(21,34,23,36)(24,38,40,48)(27,43,28,44)(29,45,32,33)(37,42,39,47);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,27,25,28)(19,29,31,32)(20,34,35,36)(24,39,40,37)(26,43,41,44)(30,45,46,33)(38,47,48,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,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, 6355751);