#6722461

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1548 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)(11,16)(12,14)(13,15)(17,18,22,27)(19,28,31,24)(20,32,35,25)(21,36,23,26)(29,39,47,45)(30,40,37,41)(33,42,46,48)(34,43,38,44);(1,2)(3,6)(4,5)(7,8)(17,19,22,31)(18,24,27,28)(20,33,35,46)(21,30,23,37)(25,42,32,48)(26,40,36,41)(29,38,47,34)(39,43,45,44);(9,11,13,12)(10,14,15,16)(17,20,22,35)(18,25,27,32)(19,29,31,47)(21,34,23,38)(24,39,28,45)(26,43,36,44)(30,46,37,33)(40,42,41,48);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,26,27,36)(19,30,31,37)(20,34,35,38)(24,40,28,41)(25,43,32,44)(29,46,47,33)(39,42,45,48);(2,5)(6,8)(9,12,13,11)(10,14,15,16)(19,30)(21,23)(24,41)(26,36)(28,40)(29,46)(31,37)(33,47)(34,38)(39,48)(42,45)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6629657);