Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | |
3478 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)(9,10,15,17,14,11,13,18)(12,16,22,23,20,19,21,24);(5,6)(7,8)(9,11,14,10)(12,19,20,16)(13,18,15,17)(21,24,22,23);(2,4)(6,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24);(2,4)(10,17)(11,18)(13,15)(16,23)(19,24)(21,22);(10,11)(12,20)(17,18)(21,22)Polycyclic presentation:
2^9|a^2=fi,b^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308183, #72308260, #72308312, … |
| maximal normal subgroup of | #72308183, #72308260, #72308312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408054);