#8691867

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

4187 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,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, 8599063);