Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×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 | |
6785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(6,8)(10,11)(13,14)(15,20)(16,18)(17,19);(1,2,3,4)(13,15)(14,16)(17,18)(19,20);(9,10,12,11)(13,16)(14,18)(15,19)(17,20);(1,2,3,4)(5,6,7,8)(9,11,12,10)(13,16)(14,15)(17,20)(18,19);(1,2)(3,4)(13,14)(15,20)(16,18)(17,19)Polycyclic presentation:
2^9|b^2=fh,c^2=g,d^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72231767, #72232287, #72232305, … |
| maximal normal subgroup of | #72231767, #72232287, #72232305, … |
| maximal quotient of | #72278348, #72279228, #72299113, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 730652);