#410441

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

799 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=f,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,h^a=hi,c^b=cg,d^b=di,e^c=ei

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, 317637);