Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | #23428 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14,18,22)(15,19,24,27)(16,20,17,21)(23,26,25,28);(6,9)(10,12)(15,23)(16,17)(19,26)(20,21)(24,25)(27,28);(2,4)(13,15)(14,19)(16,23)(17,25)(18,24)(20,26)(21,28)(22,27);(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,23)(17,25)(18,24)(20,26)(21,28)(22,27);(13,16,18,17)(14,20,22,21)(15,23,24,25)(19,26,27,28)Polycyclic presentation:
2^9|a^2=fi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72281770, #72282034, #72282387, … |
| maximal normal subgroup of | #72281770, #72282034, #72282387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346632);