#4414410

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,4)(6,8)(10,14)(11,17)(12,19)(22,24)(26,28)(31,33)(32,36)(39,40);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,26,30,28)(27,33,35,31)(29,32,38,36)(34,40,37,39);(5,6)(7,8)(9,10)(11,15)(12,20)(13,14)(16,19)(17,21)(18,24)(22,23)(25,27,30,35)(26,31,28,33)(29,37,38,34)(32,40,36,39);(1,2)(3,4)(5,6)(7,8)(9,11,13,17)(10,15,14,21)(12,18,19,23)(16,22,20,24)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(9,12,13,19)(10,16,14,20)(11,18,17,23)(15,22,21,24)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4321606);