Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
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×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1866 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)(7,12)(8,20)(9,25)(10,14)(11,15)(13,23)(16,33)(17,19)(18,29)(21,27)(22,32)(24,31)(26,28)(30,35)(34,36);(1,3,4,2)(5,7,10,18,11,19,9,26)(6,12,14,29,15,17,25,28)(8,21,24,35,22,36,23,16)(13,30,20,34,31,33,32,27);(5,8)(6,13)(7,16)(9,24)(10,23)(11,22)(12,27)(14,32)(15,31)(17,34)(18,36)(19,35)(20,25)(21,26)(28,30)(29,33);(5,6)(7,17)(8,13)(9,25)(10,14)(11,15)(12,19)(16,34)(18,28)(20,24)(21,33)(22,31)(23,32)(26,29)(27,35)(30,36)Polycyclic presentation:
2^9|a^2=h,b^2=gh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^b=fi,e^c=ei,f^c=fi,g^c=gi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214895, #72307003, #72307310, … |
| maximal normal subgroup of | #72214895, #72307003, #72307310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273524);