#8798711

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

4642 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=hi,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8705907);