Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 | #39341 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2197 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,22,24,28)(16,21,18,25)(19,26,27,23);(1,2)(3,4)(6,9)(10,12)(13,14)(15,19)(16,25)(17,20)(18,21)(22,23)(24,27)(26,28);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,23)(17,24)(18,26)(20,27)(21,28)(22,25);(1,2)(3,4)(14,20)(19,27)(21,25)(22,28);(2,4)(13,16,17,18)(14,21,20,25)(15,23,24,26)(19,28,27,22)Polycyclic presentation:
2^9|a^2=i,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72282182, #72282430 |
| maximal normal subgroup of | #72282182, #72282430 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482149);