Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2133 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)(14,19)(15,23)(16,17)(20,22)(21,25)(24,26);(1,2)(3,4)(6,9)(10,12)(14,20)(15,24)(16,17)(19,22)(21,25)(23,26);(5,7,8,11)(6,10,9,12)(13,14,17,20,18,22,16,19)(15,25,24,27,26,21,23,28);(2,4)(13,15,17,24,18,26,16,23)(14,21,20,28,22,25,19,27);(6,9)(10,12)(15,26)(21,25)(23,24)(27,28)Polycyclic presentation:
2^9|c^2=fgi,d^2=gi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257285, #72257625, #72263850, … |
| maximal normal subgroup of | #72257285, #72257625, #72263850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468361);