#8720779

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

4549 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=i,f^2=h,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, 8627975);