Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628896 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C2×2−1+6 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
16780 conjugacy classes of subgroups; only the 946 normal subgroups are shown.
This group has 16780 conjugacy classes of subgroups — too many to draw. Only its 946 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) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,11,9)(7,12,18,16)(8,13,21,19)(10,22,23,14)(15,27,29,24)(17,30,31,25)(20,32,33,26)(28,34,36,35);(5,7)(6,12)(8,15)(9,16)(10,17)(11,18)(13,24)(14,25)(19,27)(20,28)(21,29)(22,30)(23,31)(26,34)(32,35)(33,36);(5,8)(6,13)(7,15)(9,19)(10,20)(11,21)(12,24)(14,26)(16,27)(17,28)(18,29)(22,32)(23,33)(25,34)(30,35)(31,36);(2,4)(5,9,11,6)(7,16,18,12)(8,19,21,13)(10,14,23,22)(15,27,29,24)(17,25,31,30)(20,26,33,32)(28,34,36,35);(5,10,11,23)(6,14,9,22)(7,17,18,31)(8,20,21,33)(12,25,16,30)(13,26,19,32)(15,28,29,36)(24,34,27,35)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^b=gi,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72263043, #72263045, #72263055, … |
| maximal normal subgroup of | #72263043, #72263045, #72263055, … |
| maximal quotient of | #72312469 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536100);