Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14)(15,19)(16,17)(18,20)(21,22,26,30)(23,31,33,36)(24,29,25,28)(27,34,35,32);(1,2,3,4)(14,17)(18,19)(21,23,24,32,26,33,25,34)(22,27,28,31,30,35,29,36);(5,7,8,12)(6,9,10,11)(23,32)(24,25)(27,31)(28,29)(33,34)(35,36);(13,15,16,20)(14,18,17,19)(23,33)(27,35)(31,36)(32,34)Polycyclic presentation:
2^9|a^2=fi,b^2=eh,c^2=f,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=di,e^c=ei,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323946);