Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417541 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2408 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1746 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)(7,17)(8,22)(9,16)(10,14)(11,15)(12,19)(13,26)(18,28)(20,29)(21,27)(23,36)(24,32)(25,31)(30,34)(33,35);(5,7)(6,12)(8,23)(9,21)(10,19)(11,20)(13,30)(14,29)(15,27)(16,17)(18,25)(22,35)(24,34)(26,33)(28,32)(31,36);(8,24)(12,27)(13,31)(17,29)(18,33)(22,32)(23,34)(25,26);(2,4)(5,8,9,25,11,24,10,26)(6,13,14,32,15,31,16,22)(7,18,19,23,20,33,21,34)(12,28,17,30,27,35,29,36)Polycyclic presentation:
2^9|d^2=e,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,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
| maximal subgroup of | #72256491, #72256520, #72256521, … |
| maximal normal subgroup of | #72256491, #72256520, #72256521, … |
| maximal quotient of | #72314871 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325671);