#9222119

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

4175 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)(3,10)(4,9)(5,6)(7,8)(11,16)(12,15)(13,14)(17,18,24,27)(19,28,32,25)(20,22,21,31)(23,26,30,29)(33,34,40,44)(35,41,39,37)(36,46,47,42)(38,43,48,45);(2,8)(5,11)(9,10)(12,14)(17,19)(18,25)(20,29)(21,26)(22,30)(23,31)(24,32)(27,28)(33,35,40,39)(34,41,44,37)(36,43,47,45)(38,46,48,42);(1,3,7,4)(2,9,8,10)(5,12,11,14)(6,13,16,15)(17,20,24,21)(18,22,27,31)(19,29,32,26)(23,25,30,28)(33,36,40,47)(34,42,44,46)(35,45,39,43)(37,38,41,48);(1,4,7,3)(2,10,8,9)(5,12,11,14)(6,13,16,15)(17,21,24,20)(18,22,27,31)(19,26,32,29)(23,25,30,28)(33,37)(34,35)(36,38)(39,44)(40,41)(42,45)(43,46)(47,48);(1,5)(2,6)(3,12)(4,14)(7,11)(8,16)(9,13)(10,15)(17,22)(18,21)(19,30)(20,27)(23,32)(24,31)(25,26)(28,29)(33,38)(34,43)(35,42)(36,41)(37,47)(39,46)(40,48)(44,45);(1,6)(2,11)(3,13)(4,15)(5,8)(7,16)(9,14)(10,12)(17,23,24,30)(18,26,27,29)(19,31,32,22)(20,25,21,28)(33,39,40,35)(34,41,44,37)(36,43,47,45)(38,42,48,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=g,f^2=gh,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, 9129315);