Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| 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 | #23869 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,21)(17,26)(18,22)(20,24)(23,27)(25,28);(1,2)(3,4)(6,9)(10,12)(13,15,18,23)(14,19,22,27)(16,25,17,24)(20,21,28,26);(2,4)(13,15)(14,20)(16,24)(17,25)(18,23)(19,26)(21,27)(22,28);(2,4)(15,23)(19,27)(20,28)(24,25);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,21,22,26)(15,24,23,25)(19,20,27,28)Polycyclic presentation:
2^9|b^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307419, #72307528, #72307763, … |
| maximal normal subgroup of | #72307419, #72307528, #72307763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347373);