#8696730

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

4188 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603926);