#433657

order	512
nilpotency class	3
isoclinism	isoclinic class of #419839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^b=hi,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, 340853);