#8189408

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

4613 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8096604);