Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1599 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)(5,6)(7,17)(8,22)(9,15)(10,14)(11,16)(12,20)(13,24)(18,33)(19,29)(21,28)(23,27)(25,31)(26,30)(32,35)(34,36);(5,7)(6,12)(8,23,25,18)(9,19)(10,20)(11,21)(13,27,30,33)(14,28)(15,17)(16,29)(22,36,31,32)(24,35,26,34);(5,8)(6,13)(7,18)(9,24)(10,26)(11,25)(12,27)(14,22)(15,31)(16,30)(17,32)(19,34)(20,35)(21,23)(28,36)(29,33);(7,19)(9,10)(12,28)(14,15)(17,29)(18,34)(20,21)(22,31)(23,35)(24,26)(27,36)(32,33)Polycyclic presentation:
2^9|a^2=h,b^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291121);