Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(10,15)(13,16)(17,22)(18,23)(19,24)(20,21);(5,6)(7,8)(9,10,16,24,14,19,13,15)(11,18,21,22,12,17,20,23);(1,2,3,4)(6,8)(9,11,14,12)(10,17,19,18)(13,20,16,21)(15,22,24,23);(5,6)(7,8)(9,12,14,11)(10,18,19,17)(13,21,16,20)(15,23,24,22);(6,8)(11,12)(17,18)(20,21)(22,23)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72263965, #72263975, #72264026, … |
| maximal normal subgroup of | #72263965, #72263975, #72264026, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6490366);