#8767431

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

4651 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)(11,19)(12,17)(13,18)(14,15)(16,20)(21,23)(22,24)(25,26)(27,32)(28,34)(29,30)(31,33)(35,40)(36,37)(38,39);(1,2)(3,7)(4,6)(5,8)(10,15)(13,22)(16,19)(17,23)(25,27,31,28)(26,32,33,34)(29,38,35,36)(30,39,40,37);(1,2,4,6)(3,5,8,7)(11,20)(12,21)(16,19)(17,23)(26,33)(27,28)(29,35)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,20)(10,16,15,19)(12,18,21,24)(13,17,22,23)(25,28,31,27)(26,34,33,32)(29,38,35,36)(30,39,40,37);(1,2)(3,5)(4,6)(7,8)(9,12,14,21)(10,17,15,23)(11,18,20,24)(13,16,22,19)(25,29,31,35)(26,30,33,40)(27,36,28,38)(32,37,34,39);(1,3)(2,5)(4,8)(6,7)(9,13,14,22)(10,18,15,24)(11,17,20,23)(12,16,21,19)(25,30)(26,35)(27,37)(28,39)(29,33)(31,40)(32,38)(34,36)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=gh,e^2=gh,f^2=g,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, 8674627);