#8977298

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

4085 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)(3,4)(5,13)(6,8)(7,9)(10,14)(11,15)(12,16)(33,34)(35,40)(36,42)(37,38)(39,44)(41,45)(43,48)(46,47);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(17,18,22,23)(19,24,27,32)(20,25,30,29)(21,26,31,28)(33,35,39,46)(34,40,44,47)(36,43,45,37)(38,42,48,41);(2,8)(4,7)(5,14)(11,16)(18,23)(19,27)(20,30)(21,31)(33,36,39,45)(34,41,44,42)(35,37,46,43)(38,40,48,47);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,19)(18,24)(20,26)(21,29)(22,27)(23,32)(25,31)(28,30)(33,36)(34,42)(35,43)(37,46)(38,47)(39,45)(40,48)(41,44);(1,2)(3,4)(5,10)(6,8)(7,9)(11,12)(13,14)(15,16)(17,20)(18,25)(19,28)(21,24)(22,30)(23,29)(26,27)(31,32)(33,37,39,43)(34,38,44,48)(35,45,46,36)(40,41,47,42);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,21)(18,26)(19,29)(20,24)(22,31)(23,28)(25,27)(30,32)(33,38)(34,43)(35,42)(36,47)(37,44)(39,48)(40,45)(41,46)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=g,e^2=gh,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, 8884494);