#9066121

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

4182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8973317);