Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 16777216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#525 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
8354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,8)(9,10)(11,12)(15,21)(16,24)(18,22)(19,26);(2,4)(6,8)(10,12)(13,14,17,20)(15,22,21,18)(16,19,24,26)(23,28,27,25);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28);(13,15,17,21)(14,18,20,22)(16,23,24,27)(19,25,26,28);(13,16)(14,19)(15,23)(17,24)(18,25)(20,26)(21,27)(22,28)Polycyclic presentation:
2^9|b^2=i,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304734, #72304744, #72304746, … |
| maximal normal subgroup of | #72304734, #72304744, #72304746, … |
| maximal quotient of | #72304975, #72305007, #72305020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5863623);