#8434777

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

4596 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8341973);