Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394034 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,12)(7,17)(9,10)(14,29)(15,16)(18,27)(19,31)(20,21)(22,24)(25,30)(26,35)(33,34);(5,6)(7,18)(8,14)(9,15)(10,12)(11,16)(13,17)(19,32)(20,28)(21,27)(22,25)(23,30)(24,29)(26,31)(33,36)(34,35);(2,4)(5,7,11,21)(6,13,16,28)(8,19,23,33)(9,17,10,20)(12,18,15,27)(14,26,30,35)(22,34,24,31)(25,32,29,36);(5,8)(6,14)(7,19)(9,24)(10,22)(11,23)(12,25)(13,26)(15,29)(16,30)(17,31)(18,32)(20,34)(21,33)(27,36)(28,35)Polycyclic presentation:
2^9|c^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301231);