Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3481 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,4,3)(5,6)(7,9)(8,11)(10,12)(13,14)(15,23)(16,21)(17,22)(18,20)(19,24)(25,28)(26,27);(5,7,9,6)(8,11,12,10)(13,15,18,24)(14,19,20,23)(16,25,17,26)(21,27,22,28);(6,7)(10,11)(15,24)(19,23)(25,26)(27,28);(1,3,4,2)(6,7)(10,11)(14,20)(19,23)(21,22)(27,28);(5,8,9,12)(6,10,7,11)(13,16,18,17)(14,21,20,22)(15,25,24,26)(19,27,23,28);(13,17,18,16)(14,22,20,21)(15,26,24,25)(19,28,23,27)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=cg,d^b=dh152 conjugacy classes — view the full character table.
| maximal subgroup of | #72308736, #72308773, #72312321, … |
| maximal normal subgroup of | #72308736, #72308773, #72312321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533334);