Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2408 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3912 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)(6,9)(10,12)(13,14)(15,19)(16,17)(18,20)(21,22,24,26)(23,27,28,25);(5,6,8,9)(7,10,11,12)(13,15)(14,18)(16,19)(17,20)(23,28)(25,27);(7,11)(10,12)(14,16)(18,19)(21,23)(22,25)(24,28)(26,27);(1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(13,15)(14,19)(16,18)(17,20)(21,23)(22,25)(24,28)(26,27);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,19,20,18)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72334072, #72334117, #72334135, … |
| maximal normal subgroup of | #72334072, #72334117, #72334135, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3243067);