#4309085

order	512
nilpotency class	2
isoclinism	isoclinic class of #4308740
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2401 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)(11,17)(12,19)(14,20)(15,22)(25,26,29,33)(27,31,35,30)(28,32,36,39)(34,38,40,37);(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(5,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,2)(3,4)(9,11,13,17)(10,14,16,20)(12,18,19,23)(15,21,22,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,37)(36,39)(38,40);(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(25,28,29,36)(26,32,33,39)(27,34,35,40)(30,37,31,38)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=df

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, 4216281);