Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,9,11)(7,12,8,10)(13,14,16,17)(15,19,20,18);(2,4)(5,7,9,8)(6,10,11,12)(14,17)(18,19)(21,22)(23,25)(24,26)(27,28);(2,4)(5,8,9,7)(6,10,11,12)(14,17)(18,19)(22,25)(23,28)(26,27);(13,15,16,20)(14,18,17,19)(22,26)(25,27)Polycyclic presentation:
2^9|a^2=efh,b^2=ef,c^2=ef,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264301, #72264315, #72264349, … |
| maximal normal subgroup of | #72264301, #72264315, #72264349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 322636);