#8702821

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

4421 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=h,f^2=g,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, 8610017);