Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| 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 | #2408 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1672 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)(7,18)(8,20)(9,29)(10,33)(11,16)(12,17)(13,21)(14,27)(15,30)(19,31)(22,34)(23,25)(24,32)(26,36)(28,35);(5,7)(6,13)(8,24)(9,14)(10,22)(11,21)(12,23)(15,27)(16,25)(17,34)(18,33)(19,26)(20,30)(28,29)(31,35)(32,36);(5,8)(6,14)(7,19)(9,25)(10,26)(11,27)(12,28)(13,30)(15,23)(16,35)(17,36)(18,32)(20,33)(21,24)(22,29)(31,34);(2,4)(5,9,10,30,12,32,11,31)(6,15,16,29,17,19,33,24)(7,20,21,14,23,35,22,36)(8,25,26,13,28,18,27,34)Polycyclic presentation:
2^9|d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322978 |
| maximal normal subgroup of | #72322978 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325695);