#8258616

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

4169 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)(3,4,11,7)(5,8,13,15)(10,12,16,14)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32)(33,34,38,37)(35,39,41,45)(36,40,43,44)(42,46,47,48);(1,3,6,11)(2,7,9,4)(5,10,13,16)(8,14,15,12)(19,20)(21,30)(24,26)(25,29)(35,41)(36,43)(39,45)(40,44);(5,13)(8,15)(10,16)(12,14)(17,19,23,20)(18,24,22,26)(21,27,30,28)(25,31,29,32)(33,35,38,41)(34,39,37,45)(36,42,43,47)(40,46,44,48);(1,4)(2,3)(5,12)(6,7)(8,10)(9,11)(13,14)(15,16)(17,20,23,19)(18,24,22,26)(21,28,30,27)(25,31,29,32);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,16)(14,15)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,36,38,43)(34,40,37,44)(35,42,41,47)(39,46,45,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,18)(19,24,20,26)(21,29,30,25)(27,31,28,32)(33,37,38,34)(35,39,41,45)(36,44,43,40)(42,46,47,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8165812);