#8720770

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

4016 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)(3,8)(4,5)(6,11)(7,9)(10,16)(12,15)(13,14)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(34,36)(35,41)(40,44)(43,45);(1,3,7,11)(2,8,9,6)(4,14,10,12)(5,13,16,15)(18,22)(23,25)(24,29)(30,31)(33,34,38,36)(35,42,41,39)(37,40,46,44)(43,48,45,47);(2,9)(3,11)(4,10)(13,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,34,38,36)(35,42,41,39)(37,40,46,44)(43,48,45,47);(1,4,7,10)(2,5,9,16)(3,12,11,14)(6,13,8,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,38,41)(34,39,36,42)(37,45,46,43)(40,48,44,47);(1,5)(2,10)(3,13)(4,9)(6,14)(7,16)(8,12)(11,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,34)(35,42,41,39)(37,40,46,44)(43,47,45,48);(1,6,7,8)(2,11,9,3)(4,15,10,13)(5,12,16,14)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37,38,46)(34,40,36,44)(35,43,41,45)(39,47,42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=gi,e^2=g,f^2=h,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, 8627966);