Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
3806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,9)(7,8)(15,19)(18,20)(21,22,26,24)(23,28,25,27);(1,2)(3,4)(5,6,10,9)(7,12,8,11)(22,24)(27,28);(5,7,10,8)(6,11,9,12)(13,14,17,16)(15,18,19,20)(21,23,26,25)(22,27,24,28);(5,8,10,7)(6,12,9,11)(13,15)(14,18)(16,20)(17,19)(21,24,26,22)(23,28,25,27);(1,2)(3,4)(5,9,10,6)(7,12,8,11)(13,16,17,14)(15,20,19,18)(21,25,26,23)(22,28,24,27)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=ghi,d^2=g,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72232315, #72232319, #72232419, … |
| maximal normal subgroup of | #72232315, #72232319, #72232419, … |
| maximal quotient of | #72298511, #72329196, #72330823 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 823214);