Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2854 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)(11,18)(12,13)(15,22)(16,20)(19,21)(23,24);(1,2,3,4)(5,6)(7,8)(9,10,14,17)(11,15,19,24)(12,20,13,16)(18,22,21,23);(6,8)(9,11)(10,15)(12,18)(13,21)(14,19)(16,23)(17,24)(20,22);(5,6)(7,8)(9,12,14,13)(10,16,17,20)(11,18,19,21)(15,23,24,22);(6,8)(10,17)(15,24)(16,20)(22,23)Polycyclic presentation:
2^9|b^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72257316, #72257523, #72263839, … |
| maximal normal subgroup of | #72257316, #72257523, #72263839, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481167);