Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563341 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(5,6,12,15)(7,10,18,11)(8,13,19,14)(9,17,20,16);(2,3)(5,7,10,15,12,18,11,6)(8,14,9,17,19,13,20,16);(1,2,4,3)(8,19)(9,20)(13,14)(16,17);(1,3,4,2)(5,8,11,20,12,19,10,9)(6,13,7,16,15,14,18,17);(5,9,11,8,12,20,10,19)(6,14,18,16,15,13,7,17)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=h,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72217802, #72279577, #72279651 |
| maximal normal subgroup of | #72217802, #72279577, #72279651 |
| maximal quotient of | #72216143, #72216155, #72216163 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521303);