#8602806

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

4207 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8510002);