#8533627

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

4542 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440823);