Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393988 |
| 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×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1382 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)(5,6,9,14,10,12,16,23)(7,15,18,25,19,28,22,11)(8,13,20,27,21,26,30,34)(17,29,31,35,32,36,33,24);(5,7,16,18,10,19,9,22)(6,11,23,15,12,25,14,28)(8,17,30,31,21,32,20,33)(13,24,34,29,26,35,27,36);(6,12)(13,26)(14,23)(27,34);(2,4)(5,8,16,30,10,21,9,20)(6,13,23,34,12,26,14,27)(7,17,18,31,19,32,22,33)(11,24,15,29,25,35,28,36)Polycyclic presentation:
2^9|a^2=e,b^2=ef,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250492, #72252649, #72252977, … |
| maximal normal subgroup of | #72250492, #72252649, #72252977, … |
| maximal quotient of | C2×#400795 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307991);