Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622976 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #60682 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
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)(12,25,27,52)(13,30,32,58)(14,33,31,19)(17,38,39,28)(18,41,42,26)(21,43,45,65)(23,50,48,71)(24,51,49,37)(29,57,56,54)(34,59,60,46)(36,63,62,44)(40,55,53,64)(47,70,69,67)(61,68,66,72);(2,5)(6,8)(9,11,15,22)(10,16,20,35)(12,21,27,45)(13,23,32,48)(14,24,31,49)(17,34,39,60)(18,36,42,62)(19,37,33,51)(25,43,52,65)(26,44,41,63)(28,46,38,59)(29,47,56,69)(30,50,58,71)(40,61,53,66)(54,67,57,70)(55,68,64,72);(9,12,26,55,14,29,32,38,15,27,41,64,31,56,13,28)(10,17,30,54,19,40,42,52,20,39,58,57,33,53,18,25)(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);(12,27)(17,39)(21,45)(25,52)(28,38)(29,56)(34,60)(40,53)(43,65)(46,59)(47,69)(54,57)(55,64)(61,66)(67,70)(68,72)Polycyclic presentation:
2^9|a^2=i,b^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,d^b=di,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530456);