Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,24)(20,29)(21,25)(22,30)(23,26)(27,31)(28,32);(10,14)(15,16)(17,19,23,28)(18,24,26,32)(20,21,27,22)(25,31,30,29);(1,2,3,4)(6,8)(9,11,13,12)(10,15,14,16)(17,20,22,19,23,27,21,28)(18,24,25,29,26,32,30,31);(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,25,26,30)(19,27,28,20)(24,31,32,29)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=efh,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72308758, #72308775, #72308816, … |
| maximal normal subgroup of | #72308758, #72308775, #72308816, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281704);