Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23430 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2618 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)(7,10)(8,9)(11,12)(13,14)(15,24)(16,20)(17,21)(18,22)(19,23)(25,28)(26,27);(2,4)(15,25)(16,26)(17,19)(20,27)(21,23)(24,28);(6,9)(10,12)(13,15,18,16)(14,20,22,24)(17,25,19,26)(21,27,23,28);(5,7,8,11)(6,10,9,12)(13,16,18,15)(14,20,22,24)(17,26,19,25)(21,27,23,28);(13,17,18,19)(14,21,22,23)(15,25,16,26)(20,27,24,28)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=g,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 | #72282870, #72283110, #72283287, … |
| maximal normal subgroup of | #72282870, #72283110, #72283287, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356101);