#8700324

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

4463 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,21)(13,19)(14,20)(15,16)(18,22)(23,24)(25,26)(27,37)(28,39)(29,35)(30,31)(32,36)(33,40)(34,38);(2,5)(6,7)(10,16)(12,22)(17,20)(19,24)(25,27)(26,33)(28,29)(30,38)(31,39)(32,40)(34,35)(36,37);(1,2)(3,6)(4,5)(7,8)(9,11,15,14)(10,17,16,20)(12,19,22,24)(13,18,23,21)(25,28,32,38)(26,34,36,39)(27,29,40,30)(31,33,35,37);(1,3)(2,6)(4,8)(5,7)(9,12)(10,18)(11,19)(13,20)(14,24)(15,22)(16,21)(17,23)(25,29,32,30)(26,31,36,35)(27,28,40,38)(33,39,37,34);(9,13)(10,19)(11,18)(12,20)(14,21)(15,23)(16,24)(17,22)(25,30,32,29)(26,35,36,31)(27,38,40,28)(33,34,37,39);(9,14,15,11)(10,20,16,17)(12,19,22,24)(13,18,23,21)(25,31,32,35)(26,30,36,29)(27,39,40,34)(28,33,38,37)

Polycyclic presentation:

2^9|c^2=gi,d^2=h,e^2=h,f^2=gi,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, 8607520);