#8850663

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8757859);