Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| socle | C22 | C23×D8 |
| prime radical | #33610 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,20)(11,16,22,35)(12,25,29,54)(13,30,32,58)(14,33,31,19)(17,38,40,27)(18,41,42,26)(21,43,45,65)(23,50,48,71)(24,51,49,37)(28,57,56,53)(34,59,60,46)(36,63,62,44)(39,55,52,64)(47,70,69,67)(61,68,66,72);(2,5)(6,8)(9,11)(10,16)(12,21)(13,23)(14,24)(15,22)(17,34)(18,36)(19,37)(20,35)(25,43)(26,44)(27,46)(28,47)(29,45)(30,50)(31,49)(32,48)(33,51)(38,59)(39,61)(40,60)(41,63)(42,62)(52,66)(53,67)(54,65)(55,68)(56,69)(57,70)(58,71)(64,72);(9,12,26,55,14,28,32,38,15,29,41,64,31,56,13,27)(10,17,30,53,19,39,42,54,20,40,58,57,33,52,18,25)(11,21,44,68,24,47,48,59,22,45,63,72,49,69,23,46)(16,34,50,67,37,61,62,65,35,60,71,70,51,66,36,43);(1,3,4,7)(2,6,5,8);(11,22)(16,35)(21,45)(23,48)(24,49)(34,60)(36,62)(37,51)(43,65)(44,63)(46,59)(47,69)(50,71)(61,66)(67,70)(68,72)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,e^b=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530446);