Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478248 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2840 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1095 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 1095 conjugacy classes of subgroups — too many to draw. Only its 170 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)(7,11,17,25)(8,12)(9,21)(10,14)(13,22)(15,30,18,23)(16,24,33,28)(19,26)(20,27)(29,34)(31,35,32,36);(1,2)(3,4)(5,7)(6,11,14,25)(8,18)(9,16)(10,17)(12,23,27,30)(13,28,21,24)(15,20)(19,32)(22,33)(26,36,29,35)(31,34);(2,4)(5,8,10,20)(6,12,14,27)(7,15)(9,19,22,34)(11,23)(13,29,21,26)(16,32)(17,18)(24,35)(25,30)(28,36)(31,33);(5,9)(6,13,14,21)(7,16)(8,19)(10,22)(11,24,25,28)(12,26,27,29)(15,31)(17,33)(18,32)(20,34)(23,35,30,36)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=fi,d^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,h^c=hi,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72283557, #72283616, #72283795, … |
| maximal normal subgroup of | #72283557, #72283616, #72283795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385446);