Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,14)(8,10)(9,13)(11,15)(12,17)(16,20)(18,19);(6,10)(8,17)(11,14)(12,13)(15,18)(19,20);(1,3,4,2)(8,17)(10,12)(14,20)(15,18);(5,7)(6,11)(8,18)(9,16)(10,14)(12,20)(13,19)(15,17)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,e^c=ei,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72221471, C4×#23872, #72260931, … |
| maximal normal subgroup of | C4×#23872, #72260931, #72260939, … |
| maximal quotient of | #72221471, #72282948 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280872);