#9230881

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

4156 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)(3,12)(4,9)(5,14)(6,10)(7,11)(8,15)(13,16)(17,18,22,25)(19,26,20,23)(21,29,30,24)(27,31,28,32)(33,34)(35,36)(37,45)(38,41)(39,43)(40,48)(42,46)(44,47);(1,3,7,15)(2,8,11,12)(4,13,5,10)(6,14,16,9)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,35,38,43)(34,36,41,39)(37,42,48,47)(40,44,45,46);(1,4,7,5)(2,9,11,14)(3,13,15,10)(6,12,16,8)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,45)(47,48);(1,5,7,4)(2,9,11,14)(3,10,15,13)(6,12,16,8)(17,18)(19,23)(20,26)(21,29)(22,25)(24,30)(27,32)(28,31)(33,36,38,39)(34,35,41,43)(37,46,48,44)(40,47,45,42);(1,2)(3,8)(4,9)(5,14)(6,13)(7,11)(10,16)(12,15)(17,20,22,19)(18,23,25,26)(21,27,30,28)(24,32,29,31)(33,35)(34,36)(37,47)(38,43)(39,41)(40,46)(42,48)(44,45);(1,6,7,16)(2,10,11,13)(3,14,15,9)(4,12,5,8)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,37,38,48)(34,40,41,45)(35,42,43,47)(36,44,39,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=gh,e^2=hi,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, 9138077);