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 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,11,16,12,17,10,15)(7,18,21,13,8,22,20,30)(9,24,27,32,28,14,26,33)(19,34,25,36,23,31,35,29);(1,3,4,2)(5,7,12,8)(6,13,17,30)(9,19,28,23)(10,21,11,20)(14,29,24,36)(15,22,16,18)(25,27,35,26)(31,32,34,33);(2,3)(5,8,12,7)(6,13,17,30)(9,25,28,35)(10,20,11,21)(14,31,24,34)(15,22,16,18)(19,26,23,27)(29,33,36,32);(5,9)(6,14)(7,19)(8,23)(10,27)(11,26)(12,28)(13,29)(15,33)(16,32)(17,24)(18,34)(20,25)(21,35)(22,31)(30,36)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=i,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 | #72285454, #72285522, #72285577, … |
| maximal normal subgroup of | #72285454, #72285522, #72285577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363803);