Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,16,19,20);(6,9)(7,11)(14,18)(15,16)(19,20);(1,2,3,4)(13,15)(14,16)(17,19)(18,20);(6,9)(10,12);(5,7,8,11)(6,10,9,12)(13,16,17,20)(14,19,18,15)Polycyclic presentation:
2^9|a^2=i,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72307528, #72307536, #72307651, … |
| maximal normal subgroup of | #72307528, #72307536, #72307651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592744);