Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,17,22,18,21,16,23)(15,24,25,19,27,28,26,20);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,26)(17,25)(18,27)(20,21)(22,28)(23,24);(2,4)(5,8,9,7)(6,10,11,12)(13,15)(14,20)(16,26)(17,25)(18,27)(19,21)(22,24)(23,28);(14,21)(19,20)(22,23)(24,28)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72237222, #72237245, #72237249, … |
| maximal normal subgroup of | #72237222, #72237245, #72237249, … |
| maximal quotient of | #72239104, #72265339, #72265693 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265762);