Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C42 |
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 | |
6055 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)(6,7)(13,14)(15,16)(17,18)(19,20);(1,2)(3,4)(6,7)(10,12)(14,18)(16,19);(5,6,8,7)(9,10)(11,12)(13,15)(14,19)(16,18)(17,20);(5,7,8,6)(13,16,17,19)(14,15,18,20);(1,2)(3,4)(13,15)(14,19)(16,18)(17,20)Polycyclic presentation:
2^9|c^2=i,d^2=g,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 | #72232156, #72232159, #72232188, … |
| maximal normal subgroup of | #72232156, #72232159, #72232188, … |
| maximal quotient of | C2×#855329, #72330147, #72330284, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 762525);