Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2410 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)(7,15)(8,20)(9,23)(10,22)(11,14)(12,18)(13,24)(16,19)(17,28)(21,26)(25,30)(27,29)(31,33)(32,34)(35,36);(5,7,11,19)(6,12,14,29)(8,16,22,15)(9,17,26,35)(10,27,20,18)(13,28,30,36)(21,32,23,31)(24,34,25,33);(2,4)(5,8,20,6,11,22,10,14)(7,16,27,29,19,15,18,12)(9,21,24,13,26,23,25,30)(17,32,33,36,35,31,34,28);(5,9)(6,13)(7,17)(8,21)(10,25)(11,26)(12,28)(14,30)(15,31)(16,32)(18,34)(19,35)(20,24)(22,23)(27,33)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fhi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72316933 |
| maximal normal subgroup of | #72316933 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281302);