#8726383

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

4163 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=gh,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633579);