Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,8,10)(7,11,12,9)(13,14)(15,19)(16,17)(18,20)(21,22,24,28,26,30,25,29)(23,31,32,27,34,36,33,35);(2,4)(5,7,8,12)(6,9,10,11)(21,23,26,34)(22,27,30,35)(24,33,25,32)(28,31,29,36);(2,4)(14,17)(18,19)(22,28)(23,32)(24,25)(29,30)(31,36)(33,34);(13,15,16,20)(14,18,17,19)(21,24,26,25)(22,28,30,29)(23,33,34,32)(27,31,35,36)Polycyclic presentation:
2^9|a^2=ei,b^2=efh,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=di,e^c=ei,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323605);