Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C25 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×D4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
3669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13)(4,9)(5,10)(6,11)(7,12)(8,16)(14,36)(15,37)(17,38)(18,45)(19,30)(20,31)(21,32)(22,33)(23,34)(24,51)(25,35)(26,40)(27,42)(28,44)(29,43)(39,58)(41,59)(46,63)(47,55)(48,56)(49,57)(50,64)(52,60)(53,61)(54,62);(1,3,7,17)(2,8,12,28)(4,18,21,14)(5,15,22,41)(6,16,25,44)(9,29,32,26)(10,27,33,53)(11,13,35,38)(19,46,47,39)(20,43,24,40)(23,42,49,61)(30,54,55,52)(31,45,51,36)(34,37,57,59)(48,62,50,60)(56,63,64,58);(1,4,7,21)(2,9,12,32)(3,14,17,18)(5,19,22,47)(6,24,25,20)(8,26,28,29)(10,30,33,55)(11,31,35,51)(13,36,38,45)(15,39,41,46)(16,43,44,40)(23,50,49,48)(27,52,53,54)(34,56,57,64)(37,58,59,63)(42,62,61,60);(1,5)(2,10)(3,15)(4,19)(6,23)(7,22)(8,27)(9,30)(11,34)(12,33)(13,37)(14,39)(16,42)(17,41)(18,46)(20,48)(21,47)(24,50)(25,49)(26,52)(28,53)(29,54)(31,56)(32,55)(35,57)(36,58)(38,59)(40,60)(43,62)(44,61)(45,63)(51,64);(5,22)(10,33)(15,41)(19,47)(23,49)(27,53)(30,55)(34,57)(37,59)(39,46)(42,61)(48,50)(52,54)(56,64)(58,63)(60,62)Polycyclic presentation:
2^9|b^2=i,c^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ci,g^b=ghi,f^c=fh,e^d=ei,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72256594, #72302919, #72303043, … |
| maximal normal subgroup of | #72256594, #72302919, #72303043, … |
| maximal quotient of | #72256628, #72256645, #72312240, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530084);