#8745804

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

4102 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)(3,10)(4,8)(5,7)(6,9)(11,16)(12,15)(13,14)(17,18)(19,20)(21,31)(22,27)(23,25)(24,29)(26,28)(30,32)(33,34,38,42)(35,39,45,48)(36,46,44,40)(37,41,43,47);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,19,23,29)(18,24,25,20)(21,22,28,32)(26,27,31,30)(33,35,38,45)(34,39,42,48)(36,37,44,43)(40,47,46,41);(4,12)(8,15)(11,13)(14,16)(18,25)(20,24)(26,31)(27,30)(33,36,38,44)(34,40,42,46)(35,43,45,37)(39,41,48,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20,23,24)(18,19,25,29)(21,30,28,27)(22,31,32,26)(33,36)(34,40)(35,43)(37,45)(38,44)(39,41)(42,46)(47,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,21)(18,26)(19,22)(20,30)(23,28)(24,27)(25,31)(29,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,43)(44,46)(45,48);(1,5,6,3)(2,7,9,10)(4,11,12,13)(8,16,15,14)(17,22)(18,27)(19,28)(20,26)(21,29)(23,32)(24,31)(25,30)(33,37,38,43)(34,41,42,47)(35,44,45,36)(39,40,48,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,d^2=hi,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, 8653000);