#8455703

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

4482 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=gh,e^2=ghi,f^2=gh,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, 8362899);