Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2688 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(7,11)(10,12)(13,14)(15,16)(17,19)(18,20)(21,22)(23,24)(25,26)(27,28);(1,2,4,3)(13,15,17,20)(14,16,19,18)(22,26)(23,27);(5,6,9,8)(7,10,11,12)(13,14,17,19)(15,18,20,16)(21,23,25,27)(22,24,26,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,19)(17,20);(1,3,4,2)(5,8,9,6)(7,12,11,10)(13,16,17,18)(14,15,19,20)(21,24)(22,27)(23,26)(25,28)Polycyclic presentation:
2^9|b^2=fh,c^2=ghi,e^2=fghi,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 | #72263190, #72263199, #72263217 |
| maximal normal subgroup of | #72263190, #72263199, #72263217 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 822911);