#6693129

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1376 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6600325);