#9224356

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

4158 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)(3,8)(4,14)(5,6)(7,10)(9,13)(11,16)(12,15)(17,18,24,27)(19,23,26,25)(20,22,21,32)(28,31,29,30)(33,34)(35,40)(36,38)(37,44)(39,43)(41,46)(42,47)(45,48);(1,3)(2,8)(4,15)(5,11)(6,16)(7,13)(9,10)(12,14)(17,19,24,26)(18,25,27,23)(20,28,21,29)(22,30,32,31)(33,34,38,36)(35,39,43,40)(37,41,46,44)(42,47,45,48);(1,4,7,5)(2,6,10,14)(3,11,13,15)(8,12,9,16)(17,20,24,21)(18,22,27,32)(19,28,26,29)(23,31,25,30)(33,35)(34,39)(36,40)(37,42)(38,43)(41,47)(44,48)(45,46);(1,5,7,4)(2,6,10,14)(3,11,13,15)(8,16,9,12)(17,21,24,20)(18,22,27,32)(19,29,26,28)(23,31,25,30)(33,36,38,34)(35,40,43,39)(37,41,46,44)(42,47,45,48);(1,6)(2,5)(3,12)(4,10)(7,14)(8,15)(9,11)(13,16)(17,22)(18,21)(19,30)(20,27)(23,28)(24,32)(25,29)(26,31)(33,35)(34,40)(36,39)(37,45)(38,43)(41,47)(42,46)(44,48);(1,3)(2,9)(4,11)(5,15)(6,16)(7,13)(8,10)(12,14)(17,23,24,25)(18,26,27,19)(20,31,21,30)(22,29,32,28)(33,37,38,46)(34,41,36,44)(35,42,43,45)(39,47,40,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=gi,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, 9131552);