Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460131 |
| 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 | |
1038 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)(8,19,22,12)(9,23,24,13)(10,14)(11,18)(16,28,20,25)(17,26,32,29)(21,27)(30,35)(31,36)(33,34);(5,7)(6,11,14,15)(8,20)(9,17)(10,18)(12,25,19,28)(13,26,23,29)(16,22)(21,31)(24,32)(27,35,33,36)(30,34);(5,8)(6,12)(7,16,18,20)(9,21)(10,22)(11,25,15,28)(13,27)(14,19)(17,31,32,30)(23,33)(24,34)(26,36,29,35);(2,4)(5,9,10,24)(6,13,14,23)(7,17,18,32)(8,21,22,34)(11,26,15,29)(12,27,19,33)(16,30,20,31)(25,35,28,36)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,g^c=gi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285194, #72285306, #72285413, … |
| maximal normal subgroup of | #72285194, #72285306, #72285413, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368376);