#9712266

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

4278 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9619462);