#8992907

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

4055 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=hi,e^2=h,f^2=g,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, 8900103);