Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24762 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,9,12)(7,11,8,10)(15,22)(19,26)(23,25)(27,28);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(15,23)(16,17)(19,27)(20,24)(22,25)(26,28);(2,4)(13,14,18,21)(15,19,22,26)(16,24,17,20)(23,27,25,28);(2,4)(5,7,9,8)(6,11,12,10)(13,15)(14,19)(16,23)(17,25)(18,22)(20,28)(21,26)(24,27);(5,8,9,7)(6,11,12,10)(13,16,18,17)(14,20,21,24)(15,23,22,25)(19,28,26,27)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72282971, #72283158, #72283233 |
| maximal normal subgroup of | #72282971, #72283158, #72283233 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7430280);