Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1899 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,23,19,24,17,22)(15,25,27,35,28,38,33,20)(16,29,31,37,30,21,32,36)(26,40,43,34,41,39,42,44);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(16,30)(21,29)(26,41)(31,32)(34,44)(36,37)(39,40)(42,43);(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,26)(17,27)(18,33)(19,28)(21,34)(22,25)(23,38)(24,35)(29,44)(30,41)(31,42)(32,43)(36,40)(37,39);(2,4)(7,12)(9,11)(13,16,19,30)(14,21,24,29)(15,26,28,41)(17,32,18,31)(20,34,35,44)(22,37,23,36)(25,39,38,40)(27,43,33,42);(6,10)(9,11)(13,17,19,18)(14,22,24,23)(15,27,28,33)(16,31,30,32)(20,25,35,38)(21,36,29,37)(26,42,41,43)(34,40,44,39)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6969678);