Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563337 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,15)(13,17)(14,19)(16,21)(20,23)(22,24);(2,5)(6,8)(11,19)(12,21)(14,18)(15,16)(20,24)(22,23);(1,3,4,7)(2,6,5,8)(9,11,12,19,13,20,21,24)(10,14,16,18,17,22,15,23);(10,15,17,16)(11,20)(14,18,22,23)(19,24);(2,5)(6,8)(11,20)(14,22)(18,23)(19,24)Polycyclic presentation:
2^9|c^2=fg,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213590, #72219569, #72219577, … |
| maximal normal subgroup of | #72213590, #72219569, #72219577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7520675);