#9143361

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

4199 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18,23,20)(19,24,27,28)(21,25,31,30)(22,29,32,26);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,19)(18,24)(20,28)(21,26)(22,25)(23,27)(29,31)(30,32)(33,34,38,42)(35,40,45,39)(36,37,41,47)(43,48,46,44);(4,13)(8,12)(11,14)(15,16)(18,20)(19,27)(25,30)(26,29)(33,35,38,45)(34,39,42,40)(36,43,41,46)(37,44,47,48);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,20,23,18)(19,28,27,24)(21,25,31,30)(22,29,32,26)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,5,6,3)(2,7,9,10)(4,11,13,14)(8,16,12,15)(17,21)(18,25)(19,29)(20,30)(22,28)(23,31)(24,32)(26,27)(33,36)(34,37)(35,43)(38,41)(39,44)(40,48)(42,47)(45,46);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,13)(14,16)(17,22)(18,26)(19,25)(20,29)(21,24)(23,32)(27,30)(28,31)(33,37)(34,41)(35,44)(36,42)(38,47)(39,46)(40,43)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9050557);