#9684017

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

4401 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,e^2=gh,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, 9591213);