#9026846

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

4288 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8934042);