#452733

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

909 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)(9,10,14,20)(11,21,22,31)(12,26)(13,17,15,19)(16,25,33,24)(18,28)(23,32)(27,37)(29,35)(30,36)(34,39)(38,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16,20,33)(12,27,30,39)(13,24)(14,22)(15,25)(17,31,19,21)(18,34)(23,28,38,29)(26,40)(32,36)(35,37);(1,4,5,2)(3,7,8,6)(10,17)(11,22)(13,15)(16,31)(18,26)(19,20)(21,33)(23,38)(28,29)(32,34)(35,36)(37,40);(3,8)(6,7)(9,12)(10,18)(11,23)(13,29)(14,30)(15,28)(16,32)(17,26)(19,36)(20,35)(21,37)(22,38)(24,39)(25,27)(31,34)(33,40)

Polycyclic presentation:

2^9|a^2=egh,b^2=efg,c^2=efi,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, 359929);