Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682500 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2566 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)(3,6)(4,8)(5,7)(10,13)(11,15)(18,21)(19,22)(23,24);(1,3,6,2)(4,8,5,7)(9,10,12,13)(11,14,15,16);(10,13)(14,16)(17,18,22,24,20,23,19,21);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16);(1,5,6,4)(2,7,3,8)(18,23)(21,24)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72312023, #72312066 |
| maximal normal subgroup of | #72312023, #72312066 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6705424);