Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3444 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)(7,10)(8,11)(9,12)(13,14)(15,16)(17,20)(18,19)(21,22,25,27)(23,28,26,24);(2,4)(5,7,8,12)(6,9,11,10)(13,15,17,19)(14,18,20,16)(21,23,25,26)(22,24,27,28);(1,2)(3,4)(13,15)(14,18)(16,20)(17,19)(21,24,25,28)(22,26,27,23);(5,6)(7,9)(8,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,26)(27,28);(5,7)(6,10)(8,12)(9,11)(13,16)(14,19)(15,20)(17,18)(21,22)(23,28)(24,26)(25,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72327504, #72327514, #72327587, … |
| maximal normal subgroup of | #72327504, #72327514, #72327587, … |
| maximal quotient of | #72332967 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2795307);