Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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×#2403 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4632 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)(7,11)(8,9)(10,12)(13,14)(15,20)(16,18)(17,19)(21,22,26,28)(23,25,24,27);(2,4)(6,9)(10,11)(13,15,17,16)(14,18,19,20)(21,23)(22,27)(24,26)(25,28);(1,2)(3,4)(7,12)(10,11)(15,16)(18,20)(22,28)(23,24);(5,7,8,12)(6,10,9,11)(13,16,17,15)(14,18,19,20)(21,24,26,23)(22,25,28,27);(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,18)(17,19)(21,25)(22,23)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=g,b^2=fi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72300824, #72300837, #72300912, … |
| maximal normal subgroup of | #72300824, #72300837, #72300912, … |
| maximal quotient of | #72328980, #72329019, #72329949 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1481758);