Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395154 |
| 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 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1286 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)(7,12)(8,11)(9,10)(14,17)(18,19)(22,26)(23,28)(24,27);(2,4)(6,10)(11,12)(13,14,16,17)(15,19,20,18)(21,22,27,28,25,23,24,26);(5,7,9,8)(6,11,10,12)(13,15,16,20)(14,18,17,19)(21,23,24,28,25,22,27,26);(5,8,9,7)(6,11,10,12)(21,22,24,26,25,23,27,28)Polycyclic presentation:
2^9|a^2=efg,b^2=gh,c^2=fgi,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264979, #72265028, #72265088, … |
| maximal normal subgroup of | #72264979, #72265028, #72265088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318470);