#9065960

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

4196 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)(3,11)(4,14)(5,10)(6,8)(7,9)(12,15)(13,16)(17,18,23,27)(19,24,28,30)(20,22,26,32)(21,25,31,29)(34,39)(36,40)(41,47)(44,45);(1,3,7,6)(2,8,9,11)(4,12,10,16)(5,15,14,13)(17,19)(18,24)(20,29)(21,22)(23,28)(25,26)(27,30)(31,32)(33,34,38,39)(35,40,43,36)(37,41,46,47)(42,45,48,44);(2,9)(5,14)(8,11)(13,15)(19,28)(21,31)(24,30)(25,29)(33,35,38,43)(34,40,39,36)(37,42,46,48)(41,45,47,44);(1,4)(2,5)(3,12)(6,16)(7,10)(8,15)(9,14)(11,13)(17,20)(18,22)(19,29)(21,24)(23,26)(25,28)(27,32)(30,31)(33,36)(34,35)(37,45)(38,40)(39,43)(41,48)(42,47)(44,46);(1,5)(2,10)(3,13)(4,9)(6,15)(7,14)(8,12)(11,16)(17,21)(18,25)(19,22)(20,30)(23,31)(24,26)(27,29)(28,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,45)(44,47)(46,48);(1,6,7,3)(2,11,9,8)(4,12,10,16)(5,15,14,13)(17,22)(18,26)(19,21)(20,27)(23,32)(24,25)(28,31)(29,30)(33,37,38,46)(34,41,39,47)(35,42,43,48)(36,44,40,45)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8973156);