#9230890

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

4170 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)(20,21)(22,23)(24,29)(25,32)(27,31)(28,30)(33,34)(35,36)(37,45)(38,41)(39,43)(40,48)(42,46)(44,47);(1,3,7,4)(2,8,11,9)(5,12,16,14)(6,13,10,15)(18,23)(20,25)(24,26)(27,30)(33,35,38,43)(34,36,41,39)(37,42,48,47)(40,44,45,46);(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,30)(31,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,45)(47,48);(1,4,7,3)(2,9,11,8)(5,12,16,14)(6,13,10,15)(17,19)(18,24)(20,30)(21,31)(22,29)(23,26)(25,27)(28,32)(33,36,38,39)(34,35,41,43)(37,46,48,44)(40,47,45,42);(1,5,7,16)(2,10,11,6)(3,12,4,14)(8,15,9,13)(17,20)(18,21)(19,27)(22,25)(23,32)(24,28)(26,31)(29,30)(33,35)(34,36)(37,47)(38,43)(39,41)(40,46)(42,48)(44,45);(1,6,7,10)(2,5,11,16)(3,13,4,15)(8,12,9,14)(17,21)(18,25)(19,28)(20,23)(22,32)(24,30)(26,27)(29,31)(33,37,38,48)(34,40,41,45)(35,42,43,47)(36,44,39,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=gh,e^2=ghi,f^2=gh,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, 9138086);