Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3508 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)(3,4)(9,10,15,16,14,18,13,17)(11,12,20,23,19,24,21,22);(5,6)(7,8)(10,16)(11,19)(12,22)(13,15)(17,18)(23,24);(2,4)(6,8)(9,11)(10,12)(13,21)(14,19)(15,20)(16,23)(17,22)(18,24);(2,4)(9,12)(10,11)(13,23)(14,24)(15,22)(16,21)(17,20)(18,19);(9,13,14,15)(10,17,18,16)(11,20,19,21)(12,23,24,22)Polycyclic presentation:
2^9|a^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307352, #72307781, #72307900, … |
| maximal normal subgroup of | #72307352, #72307781, #72307900, … |
| maximal quotient of | #72315873 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408044);