#8664444

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

4189 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)(3,9)(4,14)(5,10)(6,7)(8,11)(12,15)(13,16)(17,18)(19,25)(20,22)(21,24)(23,27)(26,32)(28,30)(29,31)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(1,3,8,6)(2,9,11,7)(4,13,5,12)(10,15,14,16)(18,20)(23,25)(24,28)(31,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,4,8,5)(2,10,11,14)(3,12,6,13)(7,15,9,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,35,39,37)(34,41,36,40)(38,43,46,45)(42,48,44,47);(1,5,8,4)(2,10,11,14)(3,13,6,12)(7,15,9,16)(17,20,22,18)(19,25,27,23)(21,24,30,28)(26,31,29,32)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,6,8,3)(2,9,11,7)(4,13,5,12)(10,16,14,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,37,39,35)(34,40,36,41)(38,45,46,43)(42,47,44,48);(1,7,8,9)(2,6,11,3)(4,15,5,16)(10,13,14,12)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571640);