#9221170

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

4181 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=i,d^2=gh,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, 9128366);