Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417674 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,15,11,17,10,16)(7,18,21,13,22,32,28,31)(8,14,27,34,9,25,24,35)(19,36,23,29,20,33,26,30);(1,2)(3,4)(5,7)(6,13)(8,23)(9,26)(10,28)(11,22)(12,21)(14,33)(15,32)(16,18)(17,31)(19,24)(20,27)(25,36)(29,35)(30,34);(2,4)(5,8,11,9)(6,14,17,25)(7,19,22,20)(10,27,12,24)(13,29,31,30)(15,35,16,34)(18,33,32,36)(21,26,28,23);(5,9,11,8)(6,14,17,25)(7,20,22,19)(10,24,12,27)(13,30,31,29)(15,35,16,34)(18,36,32,33)(21,23,28,26)Polycyclic presentation:
2^9|a^2=ef,c^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286662, #72287067, #72287089 |
| maximal normal subgroup of | #72286662, #72287067, #72287089 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326075);