Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395174 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2535 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1278 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)(7,15)(8,20)(9,23)(10,12)(11,17)(13,22)(14,24)(16,25)(18,26)(19,27)(21,34)(28,35)(29,32)(30,36)(31,33);(2,4)(5,7)(6,11)(8,16)(9,18)(10,19)(12,26)(13,25)(14,27)(15,23)(17,24)(20,33)(21,30)(22,32)(28,36)(29,34)(31,35);(6,12)(8,21)(14,23)(16,30)(22,35)(25,36)(29,33)(31,32);(5,8,9,22,10,21,24,35)(6,13,14,20,12,28,23,34)(7,16,18,32,19,30,17,31)(11,25,27,33,26,36,15,29)Polycyclic presentation:
2^9|a^2=efh,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281398, #72281410, #72281450, … |
| maximal normal subgroup of | #72281398, #72281410, #72281450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 322164);