#8650973

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

4597 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=gi,e^2=h,f^2=hi,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, 8558169);