Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2,5,7)(3,6,4,8)(9,10,16,22,15,21,14,20)(11,23,28,17,27,49,26,42)(12,29,39,45,33,18,32,46)(13,34,38,19,37,50,36,48)(24,52,55,43,30,51,54,40)(25,41,57,69,56,68,35,47)(31,60,64,58,63,65,62,44)(53,70,61,72,59,67,66,71);(1,3,5,4)(2,6,7,8)(9,11,14,28,15,27,16,26)(10,17,20,49,21,42,22,23)(12,30,39,55,33,24,32,54)(13,35,36,25,37,57,38,56)(18,43,46,52,29,40,45,51)(19,47,34,41,48,69,50,68)(31,61,64,53,63,66,62,59)(44,71,60,67,58,72,65,70);(1,4,5,3)(2,6,7,8)(9,12,14,32,15,33,16,39)(10,18,20,45,21,29,22,46)(11,24,26,54,27,30,28,55)(13,31,36,62,37,63,38,64)(17,40,23,51,42,43,49,52)(19,44,34,65,48,58,50,60)(25,53,35,66,56,59,57,61)(41,67,47,72,68,70,69,71);(9,13,16,38,15,37,14,36)(10,19,22,50,21,48,20,34)(11,25,28,57,27,56,26,35)(12,31,39,64,33,63,32,62)(17,41,49,69,42,68,23,47)(18,44,46,60,29,58,45,65)(24,53,55,61,30,59,54,66)(40,67,52,71,43,70,51,72)Polycyclic presentation:
2^9|a^2=ef,b^2=efg,c^2=efi,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 346773);