#8944498

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

4330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=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, 8851694);