Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2410 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4097 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)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,25)(24,27)(26,28);(1,2,3,4)(6,9)(10,11)(14,18)(16,19)(21,22,24,27)(23,26,28,25);(13,15)(14,19)(16,18)(17,20)(21,23)(22,25)(24,28)(26,27);(1,2,3,4)(5,7,8,12)(6,10,9,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,16)(14,20)(15,18)(17,19)(21,22)(23,25)(24,27)(26,28)Polycyclic presentation:
2^9|b^2=i,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72304067, #72304084 |
| maximal normal subgroup of | #72304067, #72304084 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1345286);