Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432833 |
| 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 |
| prime radical | C2×#3109 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2244 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)(7,10)(8,11)(9,12)(13,14)(15,18)(16,20)(17,19)(21,24)(22,25)(23,26)(27,28);(1,2,3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,18)(16,22)(17,21)(19,24)(20,25)(23,27)(26,28);(6,11)(7,8)(10,12)(15,21)(18,24)(22,27)(25,28);(6,12)(10,11)(14,19)(18,24)(20,26)(25,28);(6,12)(10,11)(13,16,17,23)(14,20,19,26)(15,22,21,27)(18,25,24,28)Polycyclic presentation:
2^9|b^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280764, #72280904, #72282227, … |
| maximal normal subgroup of | #72280764, #72280904, #72282227, … |
| maximal quotient of | #72305063, #72306074 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343869);