Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1961 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)(13,14,19,28,18,21,17,22)(15,16,23,25,20,27,24,26);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,24)(18,20)(19,23)(21,27)(22,26)(25,28);(1,3,4,2)(6,9)(10,12)(13,16,18,27)(14,20,21,15)(17,25,19,26)(22,24,28,23);(5,7)(6,10)(8,11)(9,12)(14,21)(15,20)(22,28)(23,24);(7,11)(10,12)(13,17,18,19)(14,22,21,28)(15,23,20,24)(16,25,27,26)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ei,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
| maximal subgroup of | #72283615, #72284089, #72284610, … |
| maximal normal subgroup of | #72283615, #72284089, #72284610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6459858);