#8118661

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

4690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8025857);