Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129424 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #35149 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1823 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1823 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)(7,15,18,29)(8,12)(9,21,23,13)(10,14)(11,17)(16,31,33,30)(19,34,36,27)(20,28)(22,25)(24,35)(26,32);(1,2)(3,4)(5,7)(6,11)(8,16)(9,22)(10,18)(12,26)(13,29)(14,25)(15,21)(17,23)(19,35)(20,33)(24,28)(27,30)(31,34)(32,36);(2,4)(5,8,10,20)(6,12,14,28)(7,16,18,33)(9,19,23,36)(11,24,25,26)(13,27,21,34)(15,30,29,31)(17,32,22,35)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72235239, #72235463, #72280954, … |
| maximal normal subgroup of | #72235239, #72235463, #72280954, … |
| maximal quotient of | #72235761, #72266166, #72281376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54766);