#8718912

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

4058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8626108);