#6680355

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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)(11,13)(12,14)(15,16)(17,18,23,20)(19,29,32,25)(21,34,36,26)(22,27,24,28)(30,33,42,38)(31,39,46,43)(35,41,37,40)(44,48,45,47);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(17,19,23,32)(18,25,20,29)(21,31,36,46)(22,33,24,38)(26,39,34,43)(27,30,28,42)(35,47,37,48)(40,44,41,45);(10,14)(13,16)(17,20,23,18)(19,30,32,42)(21,26,36,34)(22,27,24,28)(25,38,29,33)(31,45,46,44)(35,41,37,40)(39,47,43,48);(2,5)(6,8)(17,21,23,36)(18,26,20,34)(19,31,32,46)(22,35,24,37)(25,39,29,43)(27,40,28,41)(30,44,42,45)(33,47,38,48);(1,3,4,7)(2,6,5,8)(17,22)(18,27)(19,32)(20,28)(21,35)(23,24)(25,29)(26,40)(31,46)(34,41)(36,37)(39,43)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6587551);