Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1108 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)(7,12)(8,18)(9,21)(10,24)(11,15)(13,16)(14,22)(17,29)(19,27)(20,26)(23,35)(25,28)(30,34)(31,32)(33,36);(5,7,11,10)(6,12,15,24)(8,16,20,25)(9,17,23,22)(13,18,28,26)(14,21,29,35)(19,32,33,30)(27,34,36,31);(5,8)(6,13)(7,16)(9,19)(10,25)(11,20)(12,26)(14,27)(15,28)(17,30)(18,24)(21,34)(22,32)(23,33)(29,36)(31,35);(5,9,10,22,11,23,7,17)(6,14,12,21,15,29,24,35)(8,19,25,32,20,33,16,30)(13,27,26,34,28,36,18,31)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292028);