#6687404

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

1691 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=fg,d^2=fi,e^2=f,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, 6594600);