#8745801

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

4026 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)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34)(35,42)(36,38)(37,41)(39,40)(43,44)(45,48)(46,47);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(34,40)(36,45)(41,42)(43,46);(2,3)(7,9)(8,10)(15,16)(20,28)(25,30)(27,29)(31,32)(33,35,39,37)(34,41,40,42)(36,46,45,43)(38,44,48,47);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20,23,28)(18,25,22,30)(19,27,21,29)(24,31,26,32)(33,36,39,45)(34,38,40,48)(35,43,37,46)(41,47,42,44);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,21,23,19)(18,24,22,26)(20,29,28,27)(25,31,30,32)(33,37,39,35)(34,42,40,41)(36,46,45,43)(38,44,48,47);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22,23,18)(19,26,21,24)(20,25,28,30)(27,31,29,32)(33,38)(34,36)(35,44)(37,47)(39,48)(40,45)(41,46)(42,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=gi,e^2=g,f^2=gh,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, 8652997);