Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393899 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2397 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1495 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)(3,4)(5,6)(7,16,18,29)(8,13)(9,14)(10,24)(11,15)(12,19,26,20)(17,30,31,36)(21,27)(22,34)(23,28)(25,32,35,33);(5,6)(7,16)(8,13)(9,14)(10,24)(11,15)(12,20)(17,30)(18,29)(19,26)(21,27)(22,34)(23,28)(25,33)(31,36)(32,35);(5,7)(6,12)(8,17)(9,20)(10,19)(11,18)(13,25)(14,29)(15,26)(16,24)(21,33)(22,32)(23,31)(27,36)(28,35)(30,34);(2,4)(5,8,9,21,11,23,10,22)(6,13,14,27,15,28,24,34)(7,17,20,33,18,31,19,32)(12,25,29,36,26,35,16,30)Polycyclic presentation:
2^9|a^2=e,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72250213, #72257191, #72257207, … |
| maximal normal subgroup of | #72250213, #72257191, #72257207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307313);