#9224293

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

4238 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)(4,9)(5,14)(6,7)(8,12)(10,16)(13,15)(17,18,23,25)(19,20,27,21)(22,24,31,32)(26,28,30,29)(34,40)(36,46)(41,43)(44,47);(2,7)(4,13)(8,11)(10,14)(17,19)(18,20)(21,25)(22,30)(23,27)(24,29)(26,31)(28,32)(33,34)(35,41)(36,42)(37,43)(38,44)(39,40)(45,46)(47,48);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,14)(15,16)(17,20,23,21)(18,19,25,27)(22,28,31,29)(24,26,32,30)(33,35,39,37)(34,41,40,43)(36,44,46,47)(38,45,48,42);(1,3)(2,8)(4,14)(5,15)(6,12)(7,11)(9,16)(10,13)(17,21,23,20)(18,19,25,27)(22,28,31,29)(24,30,32,26)(33,36)(34,42)(35,44)(37,47)(38,41)(39,46)(40,45)(43,48);(1,4,6,13)(2,9,7,15)(3,10,12,14)(5,11,16,8)(17,22)(18,24)(19,26)(20,28)(21,29)(23,31)(25,32)(27,30)(33,37,39,35)(34,43,40,41)(36,44,46,47)(38,45,48,42);(1,5)(2,10)(3,9)(4,8)(6,16)(7,14)(11,13)(12,15)(17,19)(18,21)(20,25)(22,30)(23,27)(24,28)(26,31)(29,32)(33,38)(34,44)(35,45)(36,43)(37,42)(39,48)(40,47)(41,46)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9131489);