#8581461

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

4114 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)(3,11)(4,13)(5,10)(6,7)(8,12)(9,14)(15,16)(17,18,23,20)(19,21,24,26)(22,25,30,28)(27,29,31,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(2,7)(3,12)(9,13)(10,16)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,32)(30,31)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(2,7)(4,14)(8,11)(10,16)(17,18,23,20)(19,26,24,21)(22,25,30,28)(27,32,31,29)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,13,15,9)(17,20,23,18)(19,26,24,21)(22,25,30,28)(27,29,31,32)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,47)(44,48);(1,4)(2,9)(3,10)(5,8)(6,14)(7,13)(11,15)(12,16)(17,21,23,26)(18,24,20,19)(22,29,30,32)(25,31,28,27)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,5)(2,10)(3,9)(4,8)(6,15)(7,16)(11,14)(12,13)(17,22,23,30)(18,25,20,28)(19,27,24,31)(21,29,26,32)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=hi,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8488657);