Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427387 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1056 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)(7,15,18,11)(8,19)(9,13,23,28)(10,14)(12,22)(16,24,20,29)(17,25)(21,32,34,26)(27,31)(30,36)(33,35);(5,7)(6,11)(8,20,22,16)(9,17)(10,18)(12,24,19,29)(13,27)(14,15)(21,30,34,33)(23,31)(25,28)(26,35,32,36);(5,8)(6,12,14,19)(7,16)(9,21)(10,22)(11,24,15,29)(13,26,28,32)(17,30)(18,20)(23,34)(25,35,27,36)(31,33);(2,4)(5,9,10,23)(6,13)(7,17,18,31)(8,21,22,34)(11,25)(12,26)(14,28)(15,27)(16,30,20,33)(19,32)(24,35)(29,36)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=ef,d^2=efi,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285240, #72285306, #72285464, … |
| maximal normal subgroup of | #72285240, #72285306, #72285464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344915);