Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1092 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)(3,4)(5,6,9,14,11,15,10,24)(7,12,17,27,18,29,25,28)(8,19,22,31,21,32,23,13)(16,33,20,36,34,30,35,26);(5,7)(6,12)(8,20)(9,17)(10,25)(11,18)(13,30)(14,27)(15,29)(16,23)(19,26)(21,35)(22,34)(24,28)(31,33)(32,36);(5,8,9,23,11,21,10,22)(6,13,14,32,15,31,24,19)(7,16,17,35,18,34,25,20)(12,26,27,30,29,36,28,33);(2,4)(5,7)(6,12)(8,16)(9,17)(10,25)(11,18)(13,26)(14,27)(15,29)(19,33)(20,22)(21,34)(23,35)(24,28)(30,32)(31,36)Polycyclic presentation:
2^9|a^2=f,c^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322778 |
| maximal normal subgroup of | #72322778 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332207);