Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(13,14,18,20,19,24,17,23)(15,25,28,36,29,33,27,21)(16,22,32,34,30,37,31,38)(26,39,42,44,40,43,41,35);(2,4)(5,6)(7,10)(8,9)(11,12)(14,20)(17,18)(21,33)(22,34)(23,24)(25,36)(27,28)(31,32)(35,43)(37,38)(39,44)(41,42);(6,9)(7,11)(13,15,19,29)(14,21,24,36)(16,26,30,40)(17,28,18,27)(20,33,23,25)(22,35,37,44)(31,42,32,41)(34,43,38,39);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,26)(17,31)(18,32)(19,30)(20,34)(21,35)(23,38)(24,37)(25,39)(27,41)(28,42)(29,40)(33,43)(36,44);(6,9)(10,12)(16,30)(22,37)(26,40)(31,32)(34,38)(35,44)(39,43)(41,42)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6424140);