Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2428 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,8)(9,10,11,12)(13,14,17,19,18,15,16,20);(1,2,4,3)(5,6)(7,8)(10,12)(13,15,17,20,18,14,16,19);(6,8)(10,12)(14,19)(15,20)(16,17);(1,3,4,2)(10,12)(14,15)(19,20)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=efgi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249723, #72249725, #72249752, … |
| maximal normal subgroup of | #72249723, #72249725, #72249752, … |
| maximal quotient of | #72250675, #72250723, #72322295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279873);