#452672

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

653 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=f,d^2=e,f^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=de,h^b=hi,h^d=hi

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, 359868);