#8284500

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

4265 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,9)(3,10,4,7)(5,8,13,15)(11,16,12,14)(17,18,23,26)(19,24,27,22)(20,21,25,31)(28,29,32,30)(33,34)(35,43)(36,38)(37,41)(39,40)(42,46)(44,45)(47,48);(1,3,6,4)(2,7,9,10)(5,11,13,12)(8,14,15,16)(17,19,23,27)(18,24,26,22)(20,28,25,32)(21,29,31,30)(34,40)(36,46)(41,43)(44,47);(5,13)(8,15)(11,12)(14,16)(19,27)(22,24)(28,32)(29,30)(33,35,39,37)(34,41,40,43)(36,44,46,47)(38,48,42,45);(1,2)(3,7)(4,10)(5,8)(6,9)(11,14)(12,16)(13,15)(17,20)(18,21)(19,28)(22,30)(23,25)(24,29)(26,31)(27,32)(33,36)(34,42)(35,44)(37,47)(38,40)(39,46)(41,45)(43,48);(1,4,6,3)(2,7,9,10)(5,12,13,11)(8,14,15,16)(17,21)(18,25)(19,29)(20,26)(22,28)(23,31)(24,32)(27,30)(33,37,39,35)(34,43,40,41)(36,44,46,47)(38,48,42,45);(1,5,6,13)(2,8,9,15)(3,11,4,12)(7,14,10,16)(17,22)(18,19)(20,30)(21,28)(23,24)(25,29)(26,27)(31,32)(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,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8191696);