Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498904 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2117 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 2117 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,20)(9,24)(10,12)(11,17)(13,23)(14,27)(16,33)(18,34)(19,30)(21,28)(22,31)(25,35)(26,32)(29,36);(5,7)(6,11)(8,21)(9,25)(10,19)(12,28)(13,30)(14,16)(15,20)(17,23)(18,27)(22,36)(24,29)(26,33)(31,34)(32,35);(2,4)(5,6)(7,16)(8,13)(9,24)(10,12)(11,25)(14,27)(15,33)(17,35)(18,28)(19,29)(20,23)(21,34)(22,32)(26,31)(30,36);(6,12)(11,28)(13,20)(14,24)(15,30)(16,29)(31,32)(34,35)Polycyclic presentation:
2^9|e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^c=eh,g^c=gi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72266742, #72266866, #72267431, … |
| maximal normal subgroup of | #72266742, #72266866, #72267431, … |
| maximal quotient of | #72266746, #72266756, #72266765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406104);