#8434621

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

4594 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)(12,17)(13,18)(14,19)(16,21)(20,22)(23,24)(25,26)(27,37)(28,34)(29,31)(30,39)(32,36)(33,40)(35,38);(1,2)(3,8)(4,5)(6,7)(9,11,14,21)(10,15,19,16)(12,18,22,24)(13,20,23,17)(25,27)(26,33)(28,30)(29,38)(31,39)(32,40)(34,35)(36,37);(1,3,6,4)(2,8,7,5)(9,11)(10,16)(12,18)(13,20)(14,21)(15,19)(17,23)(22,24)(25,28,32,29)(26,34,36,31)(27,30,40,38)(33,35,37,39);(1,4,6,3)(2,8,7,5)(9,11)(10,15)(12,18)(13,17)(14,21)(16,19)(20,23)(22,24)(25,29,32,28)(26,34,36,31)(27,38,40,30)(33,35,37,39);(9,12,14,22)(10,17,19,20)(11,18,21,24)(13,16,23,15)(25,30,32,38)(26,35,36,39)(27,28,40,29)(31,33,34,37);(1,5,6,8)(2,4,7,3)(9,13)(10,18)(11,20)(12,16)(14,23)(15,22)(17,21)(19,24)(25,31,32,34)(26,29,36,28)(27,39,40,35)(30,33,38,37)

Polycyclic presentation:

2^9|b^2=i,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, 8341817);