Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2057 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)(13,14,19,20,18,22,17,16)(15,21,27,23,24,28,25,26);(6,9)(7,11)(14,20)(16,22)(17,19)(21,23)(25,27)(26,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,26)(17,25)(18,24)(19,27)(20,23)(22,28);(6,9)(10,12)(13,16,19,14,18,20,17,22)(15,23,27,28,24,26,25,21);(5,7,8,11)(6,10,9,12)(15,24)(21,28)(23,26)(25,27)Polycyclic presentation:
2^9|a^2=fgi,d^2=fi,e^2=h,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=dhi,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285191, #72307895, #72308081, … |
| maximal normal subgroup of | #72285191, #72307895, #72308081, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7058304);