#8718301

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

4009 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=hi,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, 8625497);