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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2238 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)(13,14,16,19)(15,18,23,21)(17,20,25,27)(22,24,28,26);(1,2)(3,4)(6,9)(10,12)(13,15)(14,18)(16,23)(17,26)(19,21)(20,22)(24,25)(27,28);(2,4)(13,14)(15,21)(16,19)(17,20)(18,23)(22,24)(25,27)(26,28);(2,4)(5,7,8,11)(6,10,9,12)(13,14)(15,22)(16,19)(17,27)(18,26)(20,25)(21,24)(23,28);(6,9)(10,12)(14,19)(18,21)(20,27)(22,28)Polycyclic presentation:
2^9|a^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72257588, #72257677, #72263872 |
| maximal normal subgroup of | #72257588, #72257677, #72263872 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7423424);