#8193720

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

4131 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=hi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8100916);