Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500250 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24058 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #61636 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
827 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)(3,14)(4,19)(5,10)(6,12)(7,13)(8,16)(9,23)(11,25)(15,42)(17,41)(18,36)(20,50)(21,38)(22,53)(24,31)(26,35)(27,33)(28,39)(29,58)(30,45)(32,57)(34,54)(37,55)(40,44)(43,60)(46,59)(47,63)(48,51)(49,52)(56,62)(61,64);(1,3,7,18)(2,8,13,31)(4,20,21,15)(5,24,25,16)(6,17,28,47)(9,29,33,61)(10,14,11,36)(12,30,39,62)(19,48,38,60)(22,43,52,51)(23,44,27,46)(26,56,55,45)(32,59,34,40)(35,41,37,63)(42,53,50,49)(54,64,57,58);(1,4,5,22,7,21,25,52)(2,9,11,32,13,33,10,34)(3,15,16,43,18,20,24,51)(6,27,26,57,28,23,55,54)(8,29,14,40,31,61,36,59)(12,38,37,53,39,19,35,49)(17,46,45,64,47,44,56,58)(30,48,41,50,62,60,63,42);(2,10)(4,21)(5,25)(8,14)(9,32)(11,13)(12,35)(15,20)(16,24)(19,49)(23,27)(26,55)(29,59)(30,41)(31,36)(33,34)(37,39)(38,53)(40,61)(42,48)(44,46)(45,56)(50,60)(62,63)Polycyclic presentation:
2^9|b^2=i,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,f^b=fh,d^c=dh,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407448);