Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,17)(16,18)(19,20);(2,4)(5,7,9,8)(6,10,11,12)(13,15,16,19)(14,17,18,20)(21,22)(23,30)(24,28)(25,33)(26,29)(27,31)(32,35)(34,36);(2,4)(5,8,9,7)(6,10,11,12)(21,23)(22,27)(24,31)(25,34)(26,32)(28,35)(29,36)(30,33);(14,18)(17,20)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,35,36,30)Polycyclic presentation:
2^9|b^2=efh,c^2=ef,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264355, #72264365 |
| maximal normal subgroup of | #72264355, #72264365 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 314204);