Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | |
2020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,10)(8,9)(11,12);(2,3)(6,9)(10,12)(13,14,18,22)(15,19,24,28)(16,21,17,20)(23,26,25,27);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,25)(17,23)(18,24)(20,26)(21,27)(22,28);(1,3,4,2)(14,20)(16,17)(19,26)(21,22)(23,25)(27,28);(6,9)(10,12)(13,16,18,17)(14,21,22,20)(15,23,24,25)(19,26,28,27)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72257752, #72263846, #72263880, … |
| maximal normal subgroup of | #72257752, #72263846, #72263880, … |
| maximal quotient of | #72264456 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7467468);