#6718176

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1504 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(17,18,22,27)(19,28,32,24)(20,25,35,43)(21,26,37,23)(29,45,47,40)(30,38,41,31)(33,48,46,42)(34,36,44,39);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,22,32)(18,24,27,28)(20,33,35,46)(21,30,37,41)(23,38,26,31)(25,42,43,48)(29,39,47,36)(34,45,44,40);(10,15)(14,16)(17,20,22,35)(18,25,27,43)(19,29,32,47)(21,34,37,44)(23,39,26,36)(24,40,28,45)(30,42,41,48)(31,46,38,33);(2,6)(5,7)(17,21)(18,26)(19,30)(20,34)(22,37)(23,27)(24,31)(25,36)(28,38)(29,42)(32,41)(33,45)(35,44)(39,43)(40,46)(47,48);(3,8)(5,7)(9,12,13,11)(10,16,15,14)(19,31)(21,37)(23,26)(24,41)(28,30)(29,46)(32,38)(33,47)(34,44)(36,39)(40,48)(42,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,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, 6625372);