#9683682

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

4396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=gh,f^2=h,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, 9590878);