#9023549

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

4097 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,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, 8930745);