Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6)(7,8)(9,10)(11,18)(12,16)(13,15)(14,19)(17,22)(20,24)(21,23);(1,2)(3,4)(5,6)(7,8)(9,11,13,20)(10,14,15,23)(12,21,22,19)(16,18,17,24);(6,8)(10,15)(14,23)(16,17)(18,24);(2,4)(6,8)(10,15)(14,23)(16,17)(18,24);(10,16)(12,22)(14,18)(15,17)(19,21)(23,24)Polycyclic presentation:
2^9|b^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,f^e=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72282970, #72283059, #72283517, … |
| maximal normal subgroup of | #72282970, #72283059, #72283517, … |
| maximal quotient of | #72311779, #72312001 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421400);