Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3489 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4192 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,25)(15,22,27,36)(16,30,33,38)(17,24,18,23)(20,29,35,28)(21,31,39,32)(26,40,41,34)(37,42,44,43);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,26)(17,29)(18,28)(19,27)(21,34)(22,24)(23,36)(25,35)(30,44)(31,43)(32,42)(33,41)(37,38)(39,40);(7,11)(9,12)(13,16,19,33)(14,21,25,39)(15,26,27,41)(17,32,18,31)(20,34,35,40)(22,37,36,44)(23,30,24,38)(28,43,29,42);(7,11)(9,12)(13,15)(14,22)(16,26)(17,28)(18,29)(19,27)(20,24)(21,37)(23,35)(25,36)(30,40)(31,42)(32,43)(33,41)(34,38)(39,44);(6,10)(9,12)(15,27)(20,35)(22,36)(26,41)(28,29)(34,40)(37,44)(42,43)Polycyclic presentation:
2^9|a^2=i,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,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, 6377404);