Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3848 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)(7,17)(8,21)(9,12)(10,15)(11,16)(13,19)(14,22)(18,27)(20,26)(23,30)(24,29)(25,28)(31,32)(33,35)(34,36);(6,12)(9,10)(13,17)(14,21)(15,16)(19,25)(22,24)(26,28)(27,31)(29,30)(32,33)(35,36);(5,7)(6,13)(8,18)(9,19)(10,25)(11,20)(12,26)(14,27)(15,17)(16,28)(21,35)(22,32)(23,34)(24,33)(29,36)(30,31);(5,8,11,23)(6,14,16,29)(7,18,20,34)(9,24,10,22)(12,21,15,30)(13,27,28,36)(17,31,26,35)(19,33,25,32)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261238, #72261294, #72261310, … |
| maximal normal subgroup of | #72261238, #72261294, #72261310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266866);