#6718163

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1509 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)(19,29,32,24)(20,33,36,25)(21,26,23,28)(30,39,47,45)(31,43,37,40)(34,41,46,48)(35,44,38,42);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,34)(21,31)(22,32)(23,37)(25,41)(26,40)(27,29)(28,43)(30,35)(33,48)(36,46)(38,47)(39,42)(44,45);(10,15)(14,16)(17,20,22,36)(18,25,27,33)(19,30,32,47)(21,35,23,38)(24,39,29,45)(26,42,28,44)(31,46,37,34)(40,48,43,41);(2,6)(5,7)(17,21,22,23)(18,26,27,28)(19,31,32,37)(20,35,36,38)(24,40,29,43)(25,42,33,44)(30,46,47,34)(39,48,45,41);(3,8)(5,7)(9,12,13,11)(10,16,15,14)(19,31)(21,23)(24,40)(26,28)(29,43)(30,46)(32,37)(34,47)(35,38)(39,48)(41,45)(42,44)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=i,e^2=gi,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, 6625359);