Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453991 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2197 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)(5,6,10,14)(7,15,18,29)(8,12,20,28)(9,21,23,13)(11,22,26,17)(16,32,34,31)(19,30,24,27)(25,35,36,33);(5,7)(6,11)(8,16)(9,22)(10,18)(12,25)(13,29)(14,26)(15,21)(17,23)(19,35)(20,34)(24,33)(27,31)(28,36)(30,32);(2,4)(8,19)(9,23)(12,27)(13,21)(15,29)(16,33)(17,22)(20,24)(25,32)(28,30)(31,36)(34,35);(5,8)(6,12)(7,16)(9,24)(10,20)(11,25)(13,30)(14,28)(15,31)(17,35)(18,34)(19,23)(21,27)(22,33)(26,36)(29,32)Polycyclic presentation:
2^9|a^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285454, #72285522, #72285620, … |
| maximal normal subgroup of | #72285454, #72285522, #72285620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361188);