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 | |
2771 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)(8,12)(9,22)(10,14)(11,17)(13,24)(16,32)(18,31)(19,36)(20,30)(21,29)(23,28)(25,35)(26,34)(27,33);(5,7,9,17,10,18,24,35)(6,11,13,15,14,25,22,31)(8,19,20,33,21,34,23,16)(12,27,28,36,29,32,30,26);(11,25)(15,31)(26,36)(27,32);(2,4)(5,8,10,21)(6,12,14,29)(7,16,18,33)(9,23,24,20)(11,26,25,36)(13,30,22,28)(15,32,31,27)(17,34,35,19)Polycyclic presentation:
2^9|b^2=e,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 | #72256077, #72256181, #72256257, … |
| maximal normal subgroup of | #72256077, #72256181, #72256257, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324972);