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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2376 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)(7,11)(8,10)(9,12)(13,14,17,18)(15,19,23,20)(16,24,26,21)(22,25,28,27);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,17,23)(14,19,18,20)(16,25,26,27)(21,22,24,28);(2,4)(13,15)(14,20)(16,22)(17,23)(18,19)(21,25)(24,27)(26,28);(2,4)(5,7,9,8)(6,11,12,10)(13,16,17,26)(14,21,18,24)(15,22,23,28)(19,27,20,25);(5,8,9,7)(6,11,12,10)(13,14,17,18)(15,19,23,20)(16,21,26,24)(22,27,28,25)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72283141, #72283348, #72283956, … |
| maximal normal subgroup of | #72283141, #72283348, #72283956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429590);