Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34737 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,21,17,20,26,28)(15,22,24,25,23,19,27,16);(5,7)(6,9)(8,12)(10,11)(13,15,18,24,17,23,26,27)(14,19,21,16,20,22,28,25);(1,2,3,4)(7,12)(9,11)(13,16)(14,15)(17,25)(18,22)(19,26)(20,23)(21,24)(27,28);(6,10)(9,11)(14,20)(15,23)(21,28)(24,27);(15,23)(16,25)(19,22)(24,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cf,d^b=di,e^b=ei,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72312897, #72312907, #72312933 |
| maximal normal subgroup of | #72312897, #72312907, #72312933 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283335);