Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2410 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5021 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(5,6,8,7)(9,10,13,16)(11,12,15,14)(17,18)(19,21)(20,22)(23,24);(2,3)(6,7)(18,19)(20,24);(5,6,8,7)(9,11,13,15)(10,14,16,12)(17,19,21,18)(20,23,24,22);(9,10)(11,14)(12,15)(13,16);(1,3,4,2)(5,7,8,6)(9,12,13,14)(10,15,16,11)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=fg,c^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72329257, #72329270, #72329324, … |
| maximal normal subgroup of | #72329257, #72329270, #72329324, … |
| maximal quotient of | #72329868, #72332593 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 572075);