Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417859 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,15)(8,12)(9,22)(10,14)(11,17)(13,24)(16,32)(18,31)(19,29)(20,30)(21,28)(23,26)(25,34)(27,33)(35,36);(5,7,9,17,10,18,24,23)(6,11,13,15,14,26,22,31)(8,16,19,33,20,35,21,34)(12,27,28,32,30,25,29,36);(1,2,3,4)(5,8)(6,12)(7,16)(9,21)(10,20)(11,25)(13,29)(14,30)(15,32)(17,34)(18,35)(19,24)(22,28)(23,33)(26,27)(31,36);(5,7)(6,11)(8,16)(9,23)(10,18)(12,25)(13,31)(14,26)(15,22)(17,24)(19,34)(20,35)(21,33)(27,30)(28,32)(29,36)Polycyclic presentation:
2^9|b^2=e,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72255729, #72262403, #72262422 |
| maximal normal subgroup of | #72255729, #72262403, #72262422 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330115);