Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,8)(7,11)(13,14,17,20)(15,19,16,18)(22,24)(26,28);(2,3)(5,6)(7,12)(8,9)(10,11)(15,16)(18,19)(23,27)(26,28);(2,3)(6,8)(10,12)(13,15,17,16)(14,18,20,19)(21,22,25,24)(23,26,27,28);(2,3)(5,7)(6,10)(8,12)(9,11)(13,16,17,15)(14,19,20,18)(21,23,25,27)(22,26,24,28);(1,3,4,2)(5,8,9,6)(7,10,11,12)(13,14)(15,18)(16,19)(17,20)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|a^2=fh,c^2=ghi,d^2=ghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72333442, #72333464, #72333478 |
| maximal normal subgroup of | #72333442, #72333464, #72333478 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3918794);