Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3889 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(8,15)(10,16)(11,18)(14,19);(6,10)(8,15)(11,17)(12,16)(14,19)(18,20);(5,6)(7,11)(8,16)(9,12)(10,15)(13,18)(14,20)(17,19);(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,19)(16,20)Polycyclic presentation:
2^9|a^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,g^b=gi,e^c=ei,g^c=gi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214565, #72219483, #72221426, … |
| maximal normal subgroup of | #72214565, #72219483, C4×#23872, … |
| maximal quotient of | #72213382, #72213569, #72213659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266769);