Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2209 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)(7,10)(8,11)(9,12)(13,14,19,24)(15,20,28,35)(16,29,31,36)(17,23,18,22)(21,32,37,30)(25,39,42,43)(26,38,27,34)(33,40,44,41);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,25)(17,27)(18,26)(19,28)(21,33)(22,38)(23,34)(24,35)(29,39)(30,41)(31,42)(32,40)(36,43)(37,44);(7,12)(9,10)(13,16,19,31)(14,21,24,37)(15,25,28,42)(17,30,18,32)(20,33,35,44)(22,29,23,36)(26,40,27,41)(34,43,38,39);(5,7)(6,10)(8,12)(9,11)(13,17,19,18)(14,22,24,23)(15,26,28,27)(16,30,31,32)(20,34,35,38)(21,29,37,36)(25,40,42,41)(33,43,44,39);(7,12)(9,10)(13,17,19,18)(14,23,24,22)(15,27,28,26)(16,30,31,32)(20,34,35,38)(21,36,37,29)(25,41,42,40)(33,43,44,39)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=i,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6504119);