Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,11)(7,12,8,10)(14,18)(15,22)(19,25)(23,27);(5,6)(7,10)(8,12)(9,11)(14,19)(15,23)(17,20)(18,25)(21,24)(22,27);(1,2,3,4)(13,14,20,25,16,18,17,19)(15,24,27,26,22,21,23,28);(5,7,9,8)(6,10,11,12)(14,18)(19,25)(21,24)(26,28);(5,8,9,7)(6,10,11,12)(13,15,17,23,16,22,20,27)(14,21,19,26,18,24,25,28)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280952, #72281016, #72282981, … |
| maximal normal subgroup of | #72280952, #72281016, #72282981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724045);