Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394098 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
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 | |
4045 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)(8,13)(9,16)(10,14)(11,15)(12,19)(18,32)(20,30)(21,31)(22,29)(23,26)(24,28)(25,34)(27,33)(35,36);(5,7)(6,12)(8,18)(9,23)(10,19)(11,20)(13,27)(14,30)(15,26)(16,17)(21,34)(22,35)(24,33)(25,29)(28,36)(31,32);(2,4)(5,8,11,22)(6,13,15,29)(7,18,20,35)(9,24,10,21)(12,25,26,27)(14,31,16,28)(17,32,30,36)(19,34,23,33);(5,7)(6,12)(8,18)(9,19)(10,23)(11,20)(13,25)(14,17)(15,26)(16,30)(21,33)(22,35)(24,34)(27,29)(28,36)(31,32)Polycyclic presentation:
2^9|c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261238, #72287221, #72287270, … |
| maximal normal subgroup of | #72261238, #72287221, #72287270, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301296);