Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#393 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11)(8,18,20,28)(9,13)(10,14)(12,21,27,22)(15,30,19,34)(16,25)(17,26)(23,29)(24,31,35,33)(32,36);(1,2,3,4)(5,7)(6,11)(8,19,20,15)(9,17)(10,16)(12,24,27,35)(13,26)(14,25)(18,30,28,34)(21,33,22,31)(23,32)(29,36);(5,8)(6,12)(7,15)(9,22)(10,20)(11,24)(13,28)(14,27)(16,19)(17,33)(18,29)(21,23)(25,35)(26,34)(30,36)(31,32);(7,16)(11,25)(15,19)(17,32)(24,35)(26,36)(30,34)(31,33)Polycyclic presentation:
2^9|a^2=h,b^2=e,f^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=di,g^b=gi,e^c=ei,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72279381, #72279624, #72279945, … |
| maximal normal subgroup of | #72279381, #72279624, #72279945, … |
| maximal quotient of | #72281017, #72303466, C2×#357769 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264965);