Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,18,15,19,13,28)(11,16,23,32,21,31,22,38)(12,24,26,17,27,34,29,35)(20,36,37,30,25,40,39,33);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,25)(13,22)(14,23)(15,21)(17,33)(18,32)(19,31)(20,27)(24,40)(26,39)(28,38)(29,37)(30,35)(34,36);(9,12)(10,17)(11,20)(13,29)(14,26)(15,27)(16,30)(18,34)(19,35)(21,25)(22,39)(23,37)(24,28)(31,33)(32,40)(36,38);(3,8)(5,7)(9,11)(10,16)(12,20)(13,22)(14,23)(15,21)(17,30)(18,32)(19,31)(24,36)(25,27)(26,37)(28,38)(29,39)(33,35)(34,40);(2,6)(5,7)(11,21)(16,31)(20,25)(22,23)(30,33)(32,38)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,g^c=gi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72257152, #72287849 |
| maximal normal subgroup of | #72257152, #72287849 |
| maximal quotient of | #72264630, #72287386 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293274);