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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(7,11)(10,12)(13,14,17,24,19,25,18,23)(15,26,28,21,16,22,27,20);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,19)(17,27)(18,28)(21,25)(22,24)(23,26);(6,9)(10,12)(13,15,19,16)(14,21,25,20)(17,27,18,28)(22,24,26,23);(6,9)(10,12)(13,16,19,15)(14,22,25,26)(17,27,18,28)(20,23,21,24);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,25,24)(15,27,16,28)(20,26,21,22)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72256932, #72257043, #72257074, … |
| maximal normal subgroup of | #72256932, #72257043, #72257074, … |
| maximal quotient of | #72265063, #72265107, #72265690 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398700);