#8957586

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4181 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8864782);