Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2270 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)(7,10)(8,9)(11,12)(13,14)(15,22)(16,21)(17,23)(18,24)(19,25)(20,26)(27,28);(2,4)(13,15,19,16)(14,21,25,22)(17,20,27,18)(23,24,28,26);(6,9)(10,12)(13,16,20,27,19,15,18,17)(14,22,26,23,25,21,24,28);(13,17,20,16,19,27,18,15)(14,23,26,21,25,28,24,22);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,24,25,26)(15,17,16,27)(21,28,22,23)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72249920, #72250360, #72256840, … |
| maximal normal subgroup of | #72249920, #72250360, #72256840, … |
| maximal quotient of | #72264280, #72264373, #72264519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374297);