Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| 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×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1480 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)(5,6,9,11)(7,12,8,10)(13,14)(15,17)(16,18)(19,20)(22,26)(27,28);(2,4)(5,7,9,8)(6,10,11,12)(13,15,16,19)(14,17,18,20)(21,22)(23,28)(24,26)(25,27);(2,4)(22,27)(23,25)(26,28);(5,8,9,7)(6,10,11,12)(14,18)(17,20)(21,23,24,25)(22,28,26,27)Polycyclic presentation:
2^9|a^2=ei,b^2=efhi,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239610, #72239613, #72239634, … |
| maximal normal subgroup of | #72239610, #72239613, #72239634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 315444);