#8429862

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

4687 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)(3,4)(5,7)(6,8)(9,10,16,19)(11,20,22,17)(12,15,13,23)(14,24,21,18)(28,36)(32,39)(34,40)(37,38);(3,8)(4,6)(9,11)(10,17)(12,14)(13,21)(15,18)(16,22)(19,20)(23,24)(25,26)(27,31)(28,32)(29,33)(30,35)(34,38)(36,39)(37,40);(1,3,5,8)(2,6,7,4)(9,12,16,13)(10,15,19,23)(11,14,22,21)(17,18,20,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,4)(2,3)(5,6)(7,8)(9,13,16,12)(10,15,19,23)(11,21,22,14)(17,18,20,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(9,14,16,21)(10,18,19,24)(11,12,22,13)(15,20,23,17)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40);(1,2)(3,6)(4,8)(5,7)(9,15)(10,13)(11,18)(12,19)(14,20)(16,23)(17,21)(22,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,37)(36,39)(38,40)

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=g,e^2=g,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, 8337058);