#9026909

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

4272 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8934105);