#9221190

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

4172 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gi,d^2=gh,e^2=g,f^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, 9128386);