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 | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3107 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)(6,7)(10,11)(13,14,17,15)(16,19,20,18)(22,26)(24,27);(5,6,9,7)(8,11,12,10)(13,14)(15,17)(16,19)(18,20)(22,26)(23,28);(6,7)(10,11)(14,15)(18,19)(21,22,25,26)(23,27,28,24);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18)(21,23)(22,27)(24,26)(25,28);(5,8)(6,10)(7,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,24,25,27)(22,23,26,28)Polycyclic presentation:
2^9|a^2=fgh,b^2=gh,c^2=fh,d^2=f,e^2=fh,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 | #72326691, #72326696, #72326746 |
| maximal normal subgroup of | #72326691, #72326696, #72326746 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1646007);