#8493242

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

4527 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8400438);