#8434716

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

4587 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=g,e^2=gh,f^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, 8341912);