Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #93290 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1274 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | C4×M32 | C22 |
| socle | C2 | #7143 |
| prime radical | C23 | #349 |
| LCS | G ▷ C42 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,11,6,13,24,39,7,12,25,37,28,40,53,61,8,14,22,38,29,42,50,62,5,10,23,34,26,35,51,59)(3,15,17,36,19,43,45,60,20,32,46,57,48,41,52,64,16,33,44,56,27,54,55,63,18,9,21,30,47,31,49,58);(1,3)(2,9)(4,21,22,46)(5,18)(6,27)(7,20)(8,16)(10,33)(11,36,38,56)(12,15)(13,41)(14,32)(17,23,44,25)(19,29)(24,52,50,49)(26,48)(28,47)(30,37,57,34)(31,42)(35,43)(39,63,62,60)(40,54)(45,53,55,51)(58,59,64,61)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=g,f^2=i,g^2=h,h^2=i,b^a=bc,c^a=ch,e^a=efhi,f^a=fi,c^b=ci,d^b=df,g^b=gi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#94002, #72210683, #72210684, … |
| maximal normal subgroup of | C2×#94002, #72210683, #72210684, … |
| maximal quotient of | #72210603, C2×#94002 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1198);