#8508949

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

4443 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=gi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416145);