#474253

order	512
nilpotency class	3
isoclinism	isoclinic class of #455822
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

763 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)(3,6)(4,5)(7,8)(9,10,14,19)(11,16)(12,17,27,34)(13,22)(15,20,21,37)(18,32)(23,31,38,28)(24,26)(25,33)(29,36)(30,39)(35,40);(2,5)(6,8)(9,11,21,35,14,24,15,25)(10,16,20,33,19,26,37,40)(12,22,31,18,27,39,28,36)(13,23,29,34,30,38,32,17);(2,5)(3,7)(9,12,14,27)(10,17,19,34)(11,22,24,39)(13,26,30,16)(15,31,21,28)(18,35,36,25)(20,38,37,23)(29,33,32,40);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,23)(12,26)(14,30)(15,32)(16,27)(17,25)(19,36)(20,39)(21,29)(22,37)(24,38)(28,33)(31,40)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381449);