#6687811

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1693 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,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, 6595007);