#9221188

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

4157 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=i,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, 9128384);