#6690771

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#35927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1851 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)(11,14)(12,16)(13,15)(17,18,22,27)(19,28,32,24)(20,33,23,25)(21,35,37,26)(29,41,30,39)(31,40,47,45)(34,42,38,44)(36,43,46,48);(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)(21,36,37,46)(25,39,33,41)(26,43,35,48)(31,34,47,38)(40,44,45,42);(9,12,13,11)(10,14,15,16)(19,29)(20,23)(24,39)(25,33)(28,41)(30,32)(31,46)(34,38)(36,47)(40,48)(42,44)(43,45);(1,3,4,7)(2,6,5,8)(17,20,22,23)(18,25,27,33)(19,30,32,29)(21,34,37,38)(24,39,28,41)(26,42,35,44)(31,36,47,46)(40,48,45,43);(2,5)(6,8)(17,21,22,37)(18,26,27,35)(19,31,32,47)(20,34,23,38)(24,40,28,45)(25,42,33,44)(29,46,30,36)(39,48,41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6597967);