#9015268

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

4251 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)(3,4)(5,13)(6,8)(7,9)(10,14)(11,15)(12,16)(18,23)(19,25)(24,31)(26,28)(33,34,39,45)(35,40,47,48)(36,42,46,41)(37,43,44,38);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(17,18)(19,21)(20,24)(22,23)(25,30)(26,29)(27,31)(28,32)(33,35,39,47)(34,40,45,48)(36,44,46,37)(38,41,43,42);(2,8)(4,7)(5,14)(11,16)(19,25)(20,27)(21,30)(24,31)(33,36,39,46)(34,41,45,42)(35,37,47,44)(38,48,43,40);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,19)(18,21)(20,28)(22,25)(23,30)(24,32)(26,27)(29,31)(33,36)(34,42)(35,44)(37,47)(38,40)(39,46)(41,45)(43,48);(1,2)(3,4)(5,10)(6,8)(7,9)(11,12)(13,14)(15,16)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,37,39,44)(34,43,45,38)(35,46,47,36)(40,41,48,42);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,21)(18,19)(20,29)(22,30)(23,25)(24,26)(27,32)(28,31)(33,38,39,43)(34,44,45,37)(35,41,47,42)(36,40,46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=hi,e^2=i,f^2=h,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, 8922464);