Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420420 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(2,6)(7,8)(9,10,16,22)(11,17,27,42)(12,28,33,48)(13,34,36,46)(14,21,15,20)(18,32,45,31)(19,37,47,35)(23,49,51,43)(24,52,55,66)(25,41,26,40)(29,38,56,64)(30,44,61,68)(39,54,65,53)(50,63,71,72)(57,59,62,60)(58,67,70,69);(1,2,5,6)(3,7,4,8)(9,11)(10,17)(12,29)(13,24)(14,25)(15,26)(16,27)(18,43)(19,39)(20,40)(21,41)(22,42)(23,32)(28,38)(30,58)(31,51)(33,56)(34,52)(35,53)(36,55)(37,54)(44,67)(45,49)(46,66)(47,65)(48,64)(50,60)(57,63)(59,71)(61,70)(62,72)(68,69);(1,3,5,4)(2,7,6,8)(9,12,14,32,16,33,15,31)(10,18,21,28,22,45,20,48)(11,23,26,29,27,51,25,56)(13,30,35,60,36,61,37,59)(17,38,40,43,42,64,41,49)(19,44,46,57,47,68,34,62)(24,50,54,58,55,71,53,70)(39,63,52,67,65,72,66,69);(1,4,5,3)(2,7,6,8)(9,13)(10,19)(11,24)(12,30)(14,35)(15,37)(16,36)(17,39)(18,44)(20,34)(21,46)(22,47)(23,50)(25,53)(26,54)(27,55)(28,57)(29,58)(31,59)(32,60)(33,61)(38,63)(40,52)(41,66)(42,65)(43,67)(45,68)(48,62)(49,69)(51,71)(56,70)(64,72)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=eg,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 350751);