#4419038

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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

3538 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)(5,6,7,8)(11,17)(12,19)(15,20)(16,21)(25,26,30,31)(27,28,33,36)(29,37,39,32)(34,38,40,35);(10,14)(15,20)(16,21)(22,24)(26,31)(27,33)(32,37)(34,40);(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,38)(39,40);(1,2)(3,4)(5,6)(7,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28)(26,27)(29,38)(30,36)(31,33)(32,40)(34,37)(35,39);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,39)(26,32,31,37)(27,34,33,40)(28,35,36,38)

Polycyclic presentation:

2^9|a^2=fg,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=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, 4326234);