#452653

order	512
nilpotency class	3
isoclinism	isoclinic class of #417903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

810 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 359849);