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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,11)(7,12,8,10)(13,14,17,19)(15,23,25,28)(16,22,18,21)(20,26,27,24);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,19)(20,27)(21,22)(23,28);(2,4)(5,8,9,7)(6,10,11,12)(13,15,17,25)(14,20,19,27)(16,24,18,26)(21,23,22,28);(2,4);(13,16,17,18)(14,21,19,22)(15,24,25,26)(20,23,27,28)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72309029 |
| maximal normal subgroup of | #72309029 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346376);