#274213

order	512
nilpotency class	3
isoclinism	isoclinic class of #267131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181409);