#8772806

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

4265 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8680002);