#8587474

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

4406 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8494670);