Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25798 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13995 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,10,5,11)(3,13,16,28,15,8,21,23)(4,9,19,25,18,26,20,27)(14,22,31,32,30,29,17,24);(1,3)(2,8)(4,17,18,31)(5,16)(6,21)(7,15)(9,24,26,32)(10,13)(11,23)(12,28)(14,19,30,20)(22,25,29,27);(1,4)(2,9)(3,14)(5,20)(6,19)(7,18)(8,22)(10,26)(11,27)(12,25)(13,29)(15,30)(16,31)(17,21)(23,32)(24,28);(1,3)(2,8)(4,14)(5,16)(6,21)(7,15)(9,22)(10,13)(11,23)(12,28)(17,19)(18,30)(20,31)(24,25)(26,29)(27,32)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72241172, #72241179, #72241303, … |
| maximal normal subgroup of | #72241172, #72241179, #72241303, … |
| maximal quotient of | C2×#366624, #72343164 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273820);