Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417541 |
| 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 | #2598 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1325 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)(7,16,18,28)(8,20)(9,14)(10,24)(11,15)(12,19,26,23)(13,22)(17,32,33,27)(21,30)(25,29)(31,34,36,35);(2,3)(8,21)(12,26)(13,29)(16,28)(20,30)(22,25)(27,32)(31,36);(5,7)(6,12)(8,17)(9,23)(10,19)(11,18)(13,27)(14,28)(15,26)(16,24)(20,31)(21,33)(22,35)(25,34)(29,32)(30,36);(1,3,4,2)(5,8,9,22,11,21,10,25)(6,13,14,30,15,29,24,20)(7,17,23,35,18,33,19,34)(12,27,28,36,26,32,16,31)Polycyclic presentation:
2^9|a^2=e,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285592, #72285613, #72285620, … |
| maximal normal subgroup of | #72285592, #72285613, #72285620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335138);