#8750463

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

4087 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18)(19,23)(20,28)(21,25)(22,26)(24,29)(27,30)(31,32)(33,34,39,40)(35,44,38,41)(36,42,46,48)(37,45,47,43);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,25)(21,28)(22,27)(24,32)(26,30)(29,31)(34,40)(36,46)(41,44)(43,45);(2,3)(7,9)(8,10)(15,16)(19,27)(20,29)(23,30)(24,28)(33,35,39,38)(34,41,40,44)(36,43,46,45)(37,42,47,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20,22,29)(18,24,26,28)(19,25,27,31)(21,23,32,30)(33,36,39,46)(34,42,40,48)(35,45,38,43)(37,44,47,41);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,18)(19,23)(20,24)(21,31)(22,26)(25,32)(27,30)(28,29)(33,37)(34,43)(35,42)(36,44)(38,48)(39,47)(40,45)(41,46);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,32)(18,25,26,31)(19,28,27,24)(20,30,29,23)(33,38,39,35)(34,41,40,44)(36,45,46,43)(37,42,47,48)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8657659);