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 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2101 conjugacy classes of subgroups.
| 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)(7,16)(8,13)(9,22)(10,25)(11,14)(12,18)(15,27)(17,33)(19,28)(20,31)(21,34)(23,29)(24,32)(26,36)(30,35);(5,7)(6,12)(8,18)(9,23)(10,26)(11,20)(13,31)(14,29)(15,33)(16,22)(17,21)(19,27)(24,35)(25,30)(28,32)(34,36);(2,4)(5,8,11,9)(6,13,14,22)(7,17,20,35)(10,24,27,21)(12,28,29,36)(15,34,25,32)(16,33,31,30)(18,19,23,26);(6,14)(12,29)(13,22)(15,25)(16,31)(28,36)(30,33)(32,34)Polycyclic presentation:
2^9|c^2=i,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,e^c=eh,g^c=gi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72266731, #72266849, #72266866, … |
| maximal normal subgroup of | #72266731, #72266849, #72266866, … |
| maximal quotient of | #72266735, #72266765, #72266766, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406101);