Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417541 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2535 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1616 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)(7,15,18,29)(8,12)(9,13)(10,14)(11,17,24,22)(16,26,31,35)(19,32,33,25)(20,27)(21,28)(23,30)(34,36);(5,7)(6,11)(8,19)(9,22)(10,18)(12,26)(13,29)(14,24)(15,30)(16,21)(17,23)(20,33)(25,28)(27,35)(31,34)(32,36);(8,20)(11,24)(12,27)(15,29)(16,31)(19,33)(21,34)(28,36);(2,4)(5,8,9,21,10,20,23,34)(6,12,13,28,14,27,30,36)(7,16,17,19,18,31,22,33)(11,25,15,26,24,32,29,35)Polycyclic presentation:
2^9|a^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,d^c=di62 conjugacy classes — view the full character table.
| maximal subgroup of | #72256282, #72256284, #72256304, … |
| maximal normal subgroup of | #72256282, #72256284, #72256304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326978);