#8963654

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

4216 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)(4,7)(5,13)(6,8)(10,14)(11,15)(12,16)(33,34)(35,42)(36,41)(37,46)(38,43)(39,44)(40,47)(45,48);(1,3,6,4)(2,7,8,9)(5,11,14,12)(10,15,13,16)(17,18,22,20)(19,23,26,27)(21,24,29,28)(25,30,31,32)(33,35)(34,40)(36,43)(37,45)(38,46)(39,47)(41,48)(42,44);(2,8)(5,14)(7,9)(11,12)(18,20)(21,29)(23,27)(25,31)(33,36,39,37)(34,41,44,46)(35,43,47,45)(38,40,48,42);(1,4,6,3)(2,9,8,7)(5,12,14,11)(10,16,13,15)(17,19,22,26)(18,23,20,27)(21,25,29,31)(24,30,28,32)(33,37,39,36)(34,41,44,46)(35,45,47,43)(38,40,48,42);(1,2)(3,9)(4,7)(5,10)(6,8)(11,16)(12,15)(13,14)(17,20,22,18)(19,23,26,27)(21,28,29,24)(25,30,31,32)(33,35)(34,42)(36,45)(37,43)(38,46)(39,47)(40,44)(41,48);(1,5,6,14)(2,10,8,13)(3,11,4,12)(7,15,9,16)(17,21)(18,24)(19,25)(20,28)(22,29)(23,30)(26,31)(27,32)(33,38,39,48)(34,43,44,45)(35,46,47,41)(36,42,37,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8870850);