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 |
| prime radical | #42643 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1868 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)(13,14,17,23,18,22,30,20)(15,19,26,31,24,33,39,34)(16,27,29,35,28,21,36,37)(25,40,41,44,38,32,43,42);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,25)(17,26)(18,24)(20,34)(21,32)(22,33)(23,31)(27,40)(28,38)(29,41)(30,39)(35,44)(36,43)(37,42);(5,7,8,12)(6,9,10,11)(14,20)(15,24)(16,28)(17,30)(19,31)(21,37)(22,23)(27,35)(32,44)(33,34)(40,42)(41,43);(2,4)(7,12)(9,11)(13,16,18,28)(14,21,22,27)(15,25,24,38)(17,29,30,36)(19,32,33,40)(20,35,23,37)(26,41,39,43)(31,42,34,44);(6,10)(9,11)(14,22)(16,28)(19,33)(20,23)(25,38)(29,36)(31,34)(41,43)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,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, 6969617);