#8807243

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

4429 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=gh,e^2=g,f^2=gi,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, 8714439);