Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,17)(8,12)(9,19)(10,15)(11,16)(13,20)(14,18);(5,7)(6,12)(8,18)(9,16)(10,19)(11,13)(14,20)(15,17);(5,8,10,7,11,19,18,13)(6,9,14,17,16,12,15,20);(1,3,4,2)(5,9)(6,13)(7,16)(8,14)(10,17)(11,12)(15,19)(18,20)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284999, #72285483, #72285577, … |
| maximal normal subgroup of | #72284999, #72285483, #72285577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284138);