Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2513 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(7,11)(13,14,17,19)(15,22,21,18)(16,24,25,20)(23,27,26,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(14,19)(15,21)(16,25)(27,28);(2,4)(13,15,18,14,17,21,22,19)(16,26,27,24,25,23,28,20);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,23)(17,25)(18,27)(19,24)(21,26)(22,28);(2,4)(6,9)(10,12)(16,25)(20,24)(23,26)(27,28)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283018, #72283111, #72283780, … |
| maximal normal subgroup of | #72283018, #72283111, #72283780, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590930);