#9026781

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

4195 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,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, 8933977);