Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| 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×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,3)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,25)(21,23)(22,26)(24,29)(28,31)(30,32);(1,3,4,2)(9,11,12,16)(10,13,14,15)(18,23)(19,28)(20,22)(24,31)(27,32)(29,30);(2,3)(17,19,21,29)(18,24,23,27)(20,28,22,30)(25,32,26,31);(6,8)(11,16)(13,15)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,32,27,31)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264432, #72264490, #72264591, … |
| maximal normal subgroup of | #72264432, #72264490, #72264591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328916);