Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417674 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,17)(8,21)(9,16)(10,14)(11,15)(12,19)(13,24)(18,28)(20,32)(22,33)(23,31)(25,30)(26,27)(29,35)(34,36);(1,2)(3,4)(5,7)(6,12)(8,22)(9,26)(10,19)(11,20)(13,29)(14,32)(15,27)(16,17)(18,23)(21,36)(24,34)(25,35)(28,30)(31,33);(12,27)(17,32)(28,36)(29,33);(2,4)(5,8,11,25)(6,13,15,31)(7,18,20,34)(9,24,10,23)(12,28,27,36)(14,21,16,30)(17,33,32,29)(19,35,26,22)Polycyclic presentation:
2^9|d^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,d^b=de,e^b=ei,f^b=fi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72256170, #72256181, #72256289, … |
| maximal normal subgroup of | #72256170, #72256181, #72256289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324871);