Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457592 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2842 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
802 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 802 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,10,16,11,14)(7,12,25,27,19,28,18,31)(8,20,24,30,23,29,22,13)(17,32,33,36,21,35,34,26);(1,2,3,4)(5,7)(6,12)(8,21)(9,18)(10,19)(11,25)(13,26)(14,27)(15,31)(16,28)(17,23)(20,35)(22,34)(24,33)(29,32)(30,36);(2,4)(5,8,11,22,10,23,9,24)(6,13,15,20,16,30,14,29)(7,17,18,34,19,21,25,33)(12,26,27,32,28,36,31,35);(5,9,10,11)(6,14,16,15)(7,18,19,25)(8,22,23,24)(12,27,28,31)(13,29,30,20)(17,33,21,34)(26,32,36,35)Polycyclic presentation:
2^9|a^2=ef,b^2=egh,c^2=f,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=de,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379742);