Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2535 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17,21,33)(8,13)(9,23)(10,15)(11,16)(12,20,31,28)(14,27)(18,25,22,35)(19,36,30,32)(24,26)(29,34);(5,7)(6,12)(8,22)(9,25)(10,28)(11,21)(13,30)(14,32)(15,33)(16,31)(17,34)(18,24)(19,26)(20,29)(23,36)(27,35);(5,8,10,23,11,24,29,14)(6,13,15,9,16,26,34,27)(7,18,20,36,21,22,28,32)(12,30,17,35,31,19,33,25);(2,4)(5,9,10,26,11,27,29,13)(6,14,15,8,16,23,34,24)(7,19,20,25,21,30,28,35)(12,22,17,32,31,18,33,36)Polycyclic presentation:
2^9|a^2=f,c^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322924 |
| maximal normal subgroup of | #72322924 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327568);