Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #410 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×M32 | C23 |
| socle | C2 | #11494 |
| prime radical | C23 | C2×M32 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C16 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
178 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,12,6,14,26,41,7,11,25,37,27,38,53,60,8,13,23,40,28,33,46,58,4,9,21,30,22,31,50,55)(3,15,18,42,19,43,48,62,20,35,47,39,49,57,63,61,16,29,24,51,44,54,52,64,17,10,32,34,45,36,56,59);(2,9,13,11)(3,16)(5,23)(10,15,35,29)(12,30,40,37)(14,31,33,38)(17,20)(19,44)(21,25)(26,46)(34,42,39,51)(36,43,57,54)(41,55,58,60)(45,49)(50,53)(59,62,61,64);(1,3,6,19,7,20,27,49,8,16,28,44,4,17,22,45)(2,10,33,57,11,15,31,54,13,35,14,36,9,29,38,43)(5,24,26,52,25,32,53,56,23,18,46,48,21,47,50,63)(12,39,58,59,37,51,55,62,40,34,41,61,30,42,60,64)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=fg,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,c^b=ci,e^b=efi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287376, #72320907, #72343412, … |
| maximal normal subgroup of | #72287376, #72320907, #72343412, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56936);