#413462

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

944 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)(3,8)(4,6)(5,7)(9,10)(11,23)(12,27)(13,30)(14,21)(15,20)(16,22)(17,25)(18,28)(19,24)(26,36)(29,35)(31,40)(32,39)(33,37)(34,38);(1,3,5,4)(2,6,7,8)(9,11,16,26)(10,17,22,37)(12,13,29,32)(14,33,15,25)(18,30,38,39)(19,27,40,35)(20,36,21,23)(24,34,31,28);(1,4,5,3)(2,6,7,8)(9,12,16,29)(10,18,22,38)(11,24,26,31)(13,25,32,33)(14,34,15,28)(17,19,37,40)(20,35,21,27)(23,30,36,39);(9,13,15,24,16,32,14,31)(10,19,21,30,22,40,20,39)(11,12,25,28,26,29,33,34)(17,35,23,18,37,27,36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320658);