#9222121

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

4209 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:

(2,3)(5,13)(7,9)(11,14)(17,18,24,27)(19,23,26,25)(20,22,21,32)(28,31,29,30)(33,34,40,44)(35,41,39,37)(36,46,47,42)(38,43,48,45);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,16,14,15)(17,19,24,26)(18,25,27,23)(20,28,21,29)(22,30,32,31)(33,35,40,39)(34,41,44,37)(36,43,47,45)(38,46,48,42);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,15,14,16)(17,20,24,21)(18,22,27,32)(19,28,26,29)(23,31,25,30)(33,36,40,47)(34,42,44,46)(35,45,39,43)(37,38,41,48);(1,3,6,2)(4,7,12,9)(5,8,13,10)(11,16,14,15)(17,21,24,20)(18,22,27,32)(19,29,26,28)(23,31,25,30)(33,37)(34,35)(36,38)(39,44)(40,41)(42,45)(43,46)(47,48);(1,4)(2,7)(3,9)(5,14)(6,12)(8,16)(10,15)(11,13)(17,22)(18,21)(19,30)(20,27)(23,28)(24,32)(25,29)(26,31)(33,38)(34,43)(35,42)(36,41)(37,47)(39,46)(40,48)(44,45);(1,5)(2,8)(3,10)(4,11)(6,13)(7,15)(9,16)(12,14)(17,23,24,25)(18,26,27,19)(20,31,21,30)(22,29,32,28)(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=gi,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, 9129317);