Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453991 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,10,14,11,7,9,13)(8,15,18,20,19,12,17,16);(1,3,4,2)(5,7,11,6)(8,15,19,12)(9,13,10,14)(16,17,20,18);(2,3)(5,7)(6,11)(8,16)(9,13)(10,14)(12,18)(15,17)(19,20);(5,8)(6,12)(7,15)(9,18)(10,17)(11,19)(13,16)(14,20)Polycyclic presentation:
2^9|a^2=f,b^2=efi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338547, #72338549, #72338564 |
| maximal normal subgroup of | #72338547, #72338549, #72338564 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363802);