Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62295 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
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 | C22×C4×D4 |
| prime radical | #61141 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1782 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)(9,10,16,20,14,19,15,32)(11,21,26,36,23,17,25,35)(12,27,13,31,29,39,30,18)(22,37,38,33,28,40,24,34);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,28)(13,24)(14,23)(15,25)(16,26)(18,34)(19,21)(20,35)(22,29)(27,37)(30,38)(31,33)(32,36)(39,40);(2,6)(5,7)(9,12,16,13,14,29,15,30)(10,18,32,39,19,31,20,27)(11,22,26,38,23,28,25,24)(17,33,36,37,21,34,35,40);(3,8)(5,7)(11,23)(17,21)(22,28)(24,38)(25,26)(33,34)(35,36)(37,40);(2,6)(5,7)(9,13,16,29,14,30,15,12)(10,18,20,27,19,31,32,39)(11,24,26,22,23,38,25,28)(17,34,35,37,21,33,36,40)Polycyclic presentation:
2^9|a^2=h,c^2=fg,e^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^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72308458, #72308656, #72308712 |
| maximal normal subgroup of | #72308458, #72308656, #72308712 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6311152);