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 | |
3399 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)(7,11)(8,10)(9,12)(13,14,16,18)(15,17,20,19)(23,24)(26,27);(5,6,8,10)(7,9,12,11)(13,14)(15,19)(16,18)(17,20)(21,22,25,28)(23,26,24,27);(7,12)(9,11)(13,14,16,18)(15,17,20,19)(21,23,25,24)(22,26,28,27);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,22)(23,26)(24,27)(25,28);(5,6)(7,9)(8,10)(11,12)(13,15,16,20)(14,17,18,19)(21,24,25,23)(22,27,28,26)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=gh,d^2=fi,e^2=gh,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 | #72232490, #72232506, #72232542, … |
| maximal normal subgroup of | #72232490, #72232506, #72232542, … |
| maximal quotient of | #72329915, #72329931, #72332945 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1555726);