Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60399 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C2×#3484 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
7216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,17)(16,18)(19,20);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20);(8,16)(11,18)(14,20)(17,19)Polycyclic presentation:
2^9|a^2=h,b^2=fh,b^a=bf,c^a=cg,g^a=gi,g^b=gi,f^c=fi,h^c=hi,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72216064, #72219640, #72219642, … |
| maximal normal subgroup of | #72216064, #72219640, #72219642, … |
| maximal quotient of | #72235904, #72240671, #72241328, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278754);