#8726354

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

4448 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8633550);