Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,19,21)(15,24,25,27)(16,20,28,26)(17,23,18,22);(2,4)(13,15)(14,16)(17,20)(18,26)(19,25)(21,28)(22,27)(23,24);(6,8)(13,16,19,28)(14,20,21,26)(15,22,25,23)(17,27,18,24);(14,21)(15,25)(20,26)(22,23)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=gi,c^b=cg,d^b=di,e^b=ei,g^b=gi,e^c=ei,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72315669, #72315697, #72315737, … |
| maximal normal subgroup of | #72315669, #72315697, #72315737, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264714);