Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,20,18,24,17,23)(15,21,31,33,26,36,27,37)(16,28,32,38,30,22,29,34)(25,39,42,44,41,35,40,43);(5,6)(7,11)(8,10)(9,12)(14,20)(16,29)(17,19)(21,33)(23,24)(25,40)(27,31)(30,32)(34,38)(36,37)(41,42)(43,44);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,26)(14,21,24,36)(16,25,30,41)(17,27,19,31)(20,33,23,37)(22,35,28,39)(29,40,32,42)(34,43,38,44);(7,12)(9,11)(13,16,19,32,18,30,17,29)(14,22,20,34,24,28,23,38)(15,25,31,42,26,41,27,40)(21,35,33,43,36,39,37,44);(6,10)(9,11)(15,26)(16,30)(21,36)(22,28)(27,31)(29,32)(33,37)(34,38)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6928961);