Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,18,22,17,21,26,23)(15,20,25,27,24,19,28,16);(5,7,9,8)(6,10,11,12)(13,15,17,24)(14,19,21,20)(16,23,27,22)(18,25,26,28);(2,4)(13,15)(14,20)(16,23)(17,24)(18,25)(19,21)(22,27)(26,28);(5,8,9,7)(6,10,11,12)(13,16,26,19,17,27,18,20)(14,15,23,28,21,24,22,25);(15,24)(16,27)(19,20)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=fh,h^2=i,b^a=bf,c^a=cg,c^b=ci,e^b=ei,e^c=ei,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72307381, #72307430, #72307478, … |
| maximal normal subgroup of | #72307381, #72307430, #72307478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280341);