Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454095 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2208 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)(8,21)(9,25)(10,14)(11,15)(12,17)(13,22)(18,27)(19,26)(20,28)(23,29)(24,30)(31,33)(32,35)(34,36);(5,7)(6,12)(8,18)(9,20)(10,17)(11,19)(13,27)(14,26)(15,28)(16,25)(21,31)(22,32)(23,33)(24,34)(29,36)(30,35);(7,17)(8,22)(9,10)(12,26)(13,21)(14,25)(16,28)(19,20)(23,24)(27,36)(29,30)(32,33);(2,4)(5,8,11,24)(6,13,15,29)(7,18,19,34)(9,23,10,22)(12,27,28,36)(14,30,25,21)(16,31,26,35)(17,32,20,33)Polycyclic presentation:
2^9|d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72255650, #72262403, #72262421 |
| maximal normal subgroup of | #72255650, #72262403, #72262421 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361292);