Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499803 |
| 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×D4 | C24 |
| prime radical | #32899 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1951 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15)(8,19)(9,13)(10,14)(11,17)(12,21)(16,25)(18,30)(20,26)(22,34)(23,28)(24,32)(27,36)(29,31)(33,35);(1,2)(3,4)(5,7,10,18)(6,11,14,24)(8,20,22,16)(9,17,23,32)(12,26,27,25)(13,15,28,30)(19,33,34,29)(21,35,36,31);(2,4)(11,24)(13,28)(19,34)(25,26);(5,8,10,22)(6,12,14,27)(7,16,18,20)(9,21,23,36)(11,25,24,26)(13,19,28,34)(15,29,30,33)(17,31,32,35)Polycyclic presentation:
2^9|b^2=i,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fh,e^c=eh,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72303827, #72312347, #72312355, … |
| maximal normal subgroup of | #72303827, #72312347, #72312355, … |
| maximal quotient of | #72303870, #72304016, #72317797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407004);