#8463724

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

4401 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)(3,6)(4,7)(5,8)(9,10)(11,21)(12,18)(13,15)(14,24)(16,20)(17,23)(19,22)(25,26,32,30)(27,37,38,33)(28,35,29,34)(31,36,40,39);(3,8)(5,6)(9,11,16,23)(10,17,20,21)(12,22,13,14)(15,19,18,24)(25,27)(26,33)(28,36)(29,39)(30,37)(31,35)(32,38)(34,40);(1,3)(2,5)(4,8)(6,7)(9,12,16,13)(10,18,20,15)(11,22,23,14)(17,24,21,19)(25,28,32,29)(26,34,30,35)(27,36,38,39)(31,33,40,37);(1,3)(2,6)(4,8)(5,7)(9,13,16,12)(10,18,20,15)(11,14,23,22)(17,24,21,19)(25,29,32,28)(26,35,30,34)(27,36,38,39)(31,33,40,37);(9,14)(10,19)(11,12)(13,23)(15,21)(16,22)(17,18)(20,24)(25,30,32,26)(27,37,38,33)(28,35,29,34)(31,39,40,36);(1,2)(3,5)(4,7)(6,8)(9,15,16,18)(10,13,20,12)(11,19,23,24)(14,21,22,17)(25,31)(26,36)(27,35)(28,33)(29,37)(30,39)(32,40)(34,38)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8370920);