Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2304 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)(13,14,17,21)(15,22,25,19)(16,26,18,20)(23,28,24,27);(6,9)(7,11)(13,15)(14,19)(16,24)(17,25)(18,23)(20,27)(21,22)(26,28);(2,4)(15,23)(16,18)(19,27)(20,26)(22,28)(24,25);(6,9)(10,12)(13,16,17,18)(14,20,21,26)(15,24,25,23)(19,27,22,28);(5,7,8,11)(6,10,9,12)(15,24)(16,18)(19,28)(20,26)(22,27)(23,25)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257042, #72257166, #72257208, … |
| maximal normal subgroup of | #72257042, #72257166, #72257208, … |
| maximal quotient of | #72307850 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580235);