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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3403 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)(5,6,8,9)(7,11,12,10)(14,15)(18,19)(21,22,25,24)(23,26,28,27);(6,9)(7,12)(13,14,17,15)(16,19,20,18)(22,24)(26,27);(6,9)(10,11)(14,15)(18,19)(21,23,25,28)(22,26,24,27);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,23)(22,26)(24,27)(25,28);(5,7)(6,10)(8,12)(9,11)(13,16)(14,18)(15,19)(17,20)(21,24,25,22)(23,27,28,26)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=fgh,d^2=gi,e^2=fgh,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 | #72234811, #72234818, #72234824, … |
| maximal normal subgroup of | #72234811, #72234818, #72234824, … |
| maximal quotient of | #72329891, #72329931, #72333152 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1588675);