#8650352

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

4631 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=ghi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8557548);