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×#2414 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,10,11,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,24)(25,27)(26,28);(1,2)(3,4)(6,9)(10,12);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,27)(21,23)(22,28);(2,4)(13,16,18,17)(14,20,22,21)(15,24,25,26)(19,27,28,23);(13,16,18,17)(14,21,22,20)(15,24,25,26)(19,23,28,27)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=i,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 | #72283188, #72283372, #72283417, … |
| maximal normal subgroup of | #72283188, #72283372, #72283417, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345729);