Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3537 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3730 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)(14,19)(15,22)(16,18)(20,27)(23,24)(26,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,20)(16,25)(17,19)(18,21)(22,28)(23,27)(24,26);(2,4)(13,15,16,23,17,24,18,22)(14,20,25,27,19,26,21,28);(7,12)(9,11)(14,19)(20,26)(21,25)(27,28);(6,10)(9,11)(13,16,17,18)(14,21,19,25)(15,23,24,22)(20,28,26,27)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72292818, #72292824, #72292850, … |
| maximal normal subgroup of | #72292818, #72292824, #72292850, … |
| maximal quotient of | #72293220, #72293231 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300669);