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×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,20)(15,18,22,26)(16,19,24,28)(21,25,23,27);(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,21)(17,22)(19,25)(20,26)(23,24)(27,28);(1,2,3,4)(15,21)(16,24)(18,25)(19,28)(22,23)(26,27);(6,10)(9,11)(15,22)(18,26)(21,23)(25,27);(7,12)(9,11)(15,21)(16,24)(18,25)(19,28)(22,23)(26,27)Polycyclic presentation:
2^9|a^2=i,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283646, #72283706 |
| maximal normal subgroup of | #72283646, #72283706 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589417);