Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60388 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×H(7,𝔽2) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
17299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(6,10)(8,15)(11,17)(12,16)(14,19)(18,20);(5,6)(7,11)(8,16)(9,12)(10,15)(13,18)(14,20)(17,19);(8,15)(10,16)(14,19)(17,20);(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,19)(16,20);(7,13)(11,18)(14,19)(17,20)Polycyclic presentation:
2^9|a^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,g^b=gi,f^c=fi,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72227830, #72227844, #72227846, … |
| maximal normal subgroup of | #72227830, #72227844, #72227846, … |
| maximal quotient of | #72232846, #72232917, #72233035, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277943);