Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129411 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #35088 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1807 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1807 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 3 generators on 36 points:
(5,6)(7,15)(8,12)(9,21,22,13)(10,14)(11,17,24,33)(16,30)(18,29)(19,34,35,26)(20,27)(23,36,25,31)(28,32);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,25)(13,15)(14,24)(19,31)(20,32)(21,29)(22,33)(23,27)(26,30)(28,34)(35,36);(2,4)(5,8,10,20)(6,12,14,27)(7,16,18,32)(9,19,22,35)(11,23,24,25)(13,26,21,34)(15,28,29,30)(17,31,33,36)Polycyclic presentation:
2^9|a^2=g,c^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gh,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72256600, #72256601, #72256604, … |
| maximal normal subgroup of | #72256600, #72256601, #72256604, … |
| maximal quotient of | #72256710 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54764);