#424015

order	512
nilpotency class	3
isoclinism	isoclinic class of #418466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=f,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331211);