Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500797 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24260 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,11)(4,19)(5,24)(6,27)(7,13)(8,14)(9,16)(10,22)(12,25)(15,41)(17,36)(18,37)(20,50)(21,53)(23,49)(26,39)(28,58)(29,59)(30,40)(31,61)(32,45)(33,43)(34,63)(35,55)(38,46)(42,47)(44,64)(48,51)(52,56)(54,57)(60,62);(1,3,6,16,8,18,25,5)(2,9,12,11,14,24,27,37)(4,15,21,42,23,44,51,20)(7,29,28,26,30,17,46,45)(10,31,34,33,35,56,57,62)(13,39,38,59,40,32,58,36)(19,47,48,41,49,50,53,64)(22,43,54,61,55,60,63,52);(1,4)(2,10)(3,15)(5,20)(6,21)(7,22)(8,23)(9,31)(11,33)(12,34)(13,19)(14,35)(16,42)(17,43)(18,44)(24,56)(25,51)(26,52)(27,57)(28,54)(29,60)(30,55)(32,47)(36,41)(37,62)(38,48)(39,50)(40,49)(45,61)(46,63)(53,58)(59,64);(1,5,25,18,8,16,6,3)(2,11,27,9,14,37,12,24)(4,20,51,44,23,42,21,15)(7,26,46,29,30,45,28,17)(10,33,57,31,35,62,34,56)(13,36,58,32,40,59,38,39)(19,41,53,47,49,64,48,50)(22,52,63,60,55,61,54,43)Polycyclic presentation:
2^9|b^2=e,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72312286, #72312324, #72312372, … |
| maximal normal subgroup of | #72312286, #72312324, #72312372, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407995);