Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator | C22 | C23×C42 |
| socle | C23 | C24×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14,16,20,17,19,23,26)(15,18,22,25,21,24,27,28);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(15,21)(18,24)(22,27)(25,28);(7,12)(9,11)(13,15,16,22,17,21,23,27)(14,18,20,25,19,24,26,28);(6,10)(9,11);(14,19)(15,21)(20,26)(22,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,g^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cfi,e^c=ei152 conjugacy classes — view the full character table.
| maximal subgroup of | #72282815, #72282937, #72283762, … |
| maximal normal subgroup of | #72282815, #72282937, #72283762, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277499);