#8987091

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

4075 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)(3,7,10,4)(5,8,13,15)(11,14,16,12)(17,18,22,23)(19,25,27,21)(20,28,29,24)(26,31,32,30)(33,34,38,42)(35,36,40,39)(37,45,47,41)(43,48,46,44);(1,3)(2,7)(4,9)(5,11)(6,10)(8,14)(12,15)(13,16)(17,19,22,27)(18,21,23,25)(20,26,29,32)(24,31,28,30)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(3,10)(4,7)(5,13)(8,15)(18,23)(20,29)(21,25)(26,32)(33,35,38,40)(34,39,42,36)(37,46,47,43)(41,44,45,48);(1,4,6,7)(2,3,9,10)(5,14,13,12)(8,16,15,11)(17,20)(18,24)(19,26)(21,31)(22,29)(23,28)(25,30)(27,32)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,21)(18,19)(20,30)(22,25)(23,27)(24,32)(26,28)(29,31)(33,36)(34,40)(35,42)(37,44)(38,39)(41,46)(43,45)(47,48);(1,3)(2,7)(4,9)(5,11)(6,10)(8,14)(12,15)(13,16)(17,18)(19,21)(20,24)(22,23)(25,27)(26,31)(28,29)(30,32)(33,37,38,47)(34,41,42,45)(35,43,40,46)(36,44,39,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=ghi,f^2=gh,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, 8894287);