Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417682 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2459 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)(7,18)(8,13)(9,16)(10,15)(11,17)(12,21)(14,25)(19,31)(20,29)(22,35)(23,28)(24,32)(26,33)(27,30)(34,36);(1,2)(3,4)(5,7)(6,12)(8,23)(9,27)(10,21)(11,22)(13,31)(14,34)(15,35)(16,18)(17,30)(19,25)(20,24)(26,36)(28,32)(29,33);(5,8)(6,13)(7,19)(9,24)(10,25)(11,26)(12,28)(14,15)(16,32)(17,33)(18,29)(20,22)(21,36)(23,27)(30,34)(31,35);(2,4)(5,8)(6,14)(7,20)(9,25)(10,24)(11,26)(12,29)(13,15)(16,33)(17,32)(18,28)(19,22)(21,36)(23,27)(30,31)(34,35)Polycyclic presentation:
2^9|e^2=i,g^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,e^b=ei,f^b=fi,g^b=gi,g^c=gi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72255821, #72256282, #72256291 |
| maximal normal subgroup of | #72255821, #72256282, #72256291 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324878);