#8421767

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

4462 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8328963);