#453892

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

587 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)(9,10,13,16)(11,15,20,30)(12,22,24,36)(14,27,26,18)(17,29,33,25)(19,37,23,40)(21,35,28,32)(31,39,34,38);(1,3,5,8)(2,6,4,7)(9,11,14,28,13,20,26,21)(10,15,18,35,16,30,27,32)(12,23,25,38,24,19,29,39)(17,34,22,37,33,31,36,40);(1,2,5,4)(3,6,8,7)(10,16)(14,26)(15,30)(17,33)(21,28)(25,29)(31,34)(38,39);(1,4,5,2)(3,6,8,7)(9,12)(10,17)(11,19,20,23)(13,24)(14,29)(15,31,30,34)(16,33)(18,36)(21,39,28,38)(22,27)(25,26)(32,40,35,37)

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,c^2=efgi,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,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, 361088);