Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60750 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,34)(35,36);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23,25,31)(22,26,28,34)(24,30,32,36)(27,33,29,35);(7,14)(10,13)(15,20)(17,19)(22,27)(24,32)(26,33)(28,29)(30,36)(34,35);(6,11)(10,13)(12,18)(17,19);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)(22,28)(26,34)(27,29)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=cf,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304664, #72304837, #72310179, … |
| maximal normal subgroup of | #72304664, #72304837, #72310179, … |
| maximal quotient of | #72305096, #72310211 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281862);