Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2408 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,11,16)(7,17,23,31)(8,24,20,33)(9,28,29,36)(10,30,18,15)(12,22,32,21)(13,26,34,27)(14,19,35,25);(1,3,4,2)(5,7,18,22,11,23,10,21)(6,12,30,31,16,32,15,17)(8,25,27,9,20,19,26,29)(13,14,24,36,34,35,33,28);(5,8)(6,13)(7,19)(9,21)(10,26)(11,20)(12,14)(15,33)(16,34)(17,28)(18,27)(22,29)(23,25)(24,30)(31,36)(32,35);(5,9,10,25,11,29,18,19)(6,14,15,28,16,35,30,36)(7,20,21,27,23,8,22,26)(12,24,17,13,32,33,31,34)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214895, #72250063, #72250068, … |
| maximal normal subgroup of | #72214895, #72250063, #72250068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272273);