Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417704 |
| 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 | |
2262 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)(5,6,10,15,9,12,11,16)(7,17,18,26,21,29,20,13)(8,22,24,14,23,30,25,27)(19,33,34,31,35,28,32,36);(6,12)(7,18,21,20)(8,23)(13,17,26,29)(15,16)(19,32,35,34)(24,25)(28,31,33,36);(2,4)(5,7)(6,13)(8,19)(9,21)(10,20)(11,18)(12,26)(14,28)(15,29)(16,17)(22,36)(23,35)(24,32)(25,34)(27,33)(30,31);(5,8,9,23)(6,14,12,27)(7,19,21,35)(10,25,11,24)(13,28,26,33)(15,22,16,30)(17,31,29,36)(18,32,20,34)Polycyclic presentation:
2^9|a^2=f,b^2=ei,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286449, #72286995, #72287067, … |
| maximal normal subgroup of | #72286449, #72286995, #72287067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326256);