Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(10,15)(11,18)(14,17)(16,23)(19,20)(21,24);(2,4)(5,6)(7,8)(9,10)(11,19)(12,14)(13,15)(16,21)(17,22)(18,23)(20,24);(6,8)(9,11,14,18,13,21,17,24)(10,16,12,20,15,19,22,23);(6,8)(9,12)(10,17)(11,20)(13,22)(14,15)(16,24)(18,19)(21,23);(10,15)(12,22)(16,19)(20,23)Polycyclic presentation:
2^9|a^2=fi,c^2=g,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72307315, #72307607, #72307680 |
| maximal normal subgroup of | #72307315, #72307607, #72307680 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350117);