#9143364

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

4187 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=gh,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, 9050560);