#8695896

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

4633 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)(6,7)(9,10)(11,14)(12,18)(13,23)(15,17)(16,19)(20,21)(22,24)(25,26,31,29)(27,32,38,36)(28,39,37,33)(30,34,35,40);(1,2)(3,6)(4,5)(7,8)(9,11,16,15)(10,17,19,14)(12,20,13,22)(18,24,23,21)(25,27,31,38)(26,32,29,36)(28,30,37,35)(33,40,39,34);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,18,19,23)(11,20,15,22)(14,21,17,24)(25,28)(26,33)(27,35)(29,39)(30,38)(31,37)(32,34)(36,40);(1,2)(3,7)(4,5)(6,8)(9,13,16,12)(10,18,19,23)(11,20,15,22)(14,24,17,21)(25,29,31,26)(27,36,38,32)(28,39,37,33)(30,34,35,40);(1,2)(3,7)(4,5)(6,8)(9,14)(10,11)(12,21)(13,24)(15,19)(16,17)(18,20)(22,23)(25,28)(26,33)(27,30)(29,39)(31,37)(32,40)(34,36)(35,38);(1,3,4,8)(2,6,5,7)(9,15,16,11)(10,17,19,14)(12,22,13,20)(18,24,23,21)(25,30)(26,34)(27,37)(28,38)(29,40)(31,35)(32,33)(36,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603092);