Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4878 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)(9,10,12,16,14,17,13,18)(11,19,22,15,23,24,21,20);(2,4)(5,6)(7,8)(9,10)(11,20)(12,18)(13,16)(14,17)(15,23)(19,21)(22,24);(6,8)(9,11,14,23)(10,15,17,20)(12,21,13,22)(16,19,18,24);(6,8)(9,12,14,13)(10,16,17,18)(11,21,23,22)(15,19,20,24);(10,17)(15,20)(16,18)(19,24)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72308260, #72308343, #72312833, … |
| maximal normal subgroup of | #72308260, #72308343, #72312833, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345068);