#9684372

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

4506 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)(11,16)(12,18)(13,19)(14,24)(15,17)(20,21)(22,23)(26,31)(27,33)(29,36)(34,39);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,17,24)(12,20,19,22)(13,23,18,21)(25,26,30,31)(27,32,33,38)(28,36,37,29)(34,40,39,35);(1,3,5,6)(2,4,7,8)(10,17)(12,19)(16,24)(20,22)(25,27,30,33)(26,32,31,38)(28,34,37,39)(29,35,36,40);(9,12,15,19)(10,18,17,13)(11,20,14,22)(16,21,24,23)(25,27)(26,32)(28,34)(29,35)(30,33)(31,38)(36,40)(37,39);(1,2)(3,4)(5,7)(6,8)(9,13,15,18)(10,19,17,12)(11,21,14,23)(16,20,24,22)(25,28,30,37)(26,29,31,36)(27,34,33,39)(32,35,38,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,16,17,24)(12,22,19,20)(13,23,18,21)(25,29)(26,28)(27,35)(30,36)(31,37)(32,34)(33,40)(38,39)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=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, 9591568);