Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1719 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,22)(9,15)(10,13)(11,16)(12,18)(14,23)(19,27)(20,26)(21,28)(24,29)(25,30)(31,33)(32,36)(34,35);(1,3,4,2)(5,7)(6,12)(8,19)(9,20)(10,18)(11,21)(13,28)(14,27)(15,17)(16,26)(22,31)(23,32)(24,34)(25,33)(29,36)(30,35);(6,13,16,15)(7,18,21,20)(12,26)(14,29,30,22)(17,28)(19,32,34,33)(27,35)(31,36);(5,8,11,24)(6,14,16,30)(7,19,21,34)(9,25,10,23)(12,27,26,35)(13,29,15,22)(17,31,28,36)(18,32,20,33)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fgi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72317478 |
| maximal normal subgroup of | #72317478 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276075);