Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417597 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2827 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)(8,13)(9,14)(10,24)(11,15)(12,19)(17,28)(18,33)(20,26)(21,29)(22,36)(23,32)(25,31)(27,35)(30,34);(5,7,10,19,11,20,8,17)(6,12,13,16,15,28,24,26)(9,22,21,34,23,35,25,18)(14,30,31,36,32,33,29,27);(5,8,11,10)(6,13,15,24)(7,17,20,19)(9,21,23,25)(12,26,28,16)(14,29,32,31)(18,22,34,35)(27,30,36,33);(2,4)(5,9,11,23)(6,14,15,32)(7,18,20,34)(8,21,10,25)(12,27,28,36)(13,29,24,31)(16,33,26,30)(17,22,19,35)Polycyclic presentation:
2^9|b^2=e,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72251316, #72256077, #72256186, … |
| maximal normal subgroup of | #72251316, #72256077, #72256186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324973);