#8383786

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

4239 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=hi,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8290982);