#286248

order	512
nilpotency class	3
isoclinism	isoclinic class of #282043
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	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

619 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)(3,6)(4,5)(7,8)(9,10,13,16,14,18,12,17)(11,15,21,23,22,24,20,19)(25,26,29,32,30,33,28,38)(27,31,36,39,37,34,35,40);(2,5)(6,8)(9,11,12,20,14,22,13,21)(10,15,17,19,18,24,16,23)(25,27,29,36,30,37,28,35)(26,31,32,39,33,34,38,40);(1,3,4,7)(2,6,5,8)(9,10,14,18)(11,19,22,23)(12,16,13,17)(15,21,24,20)(25,26)(27,34)(28,32)(29,38)(30,33)(31,37)(35,40)(36,39);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)(25,28,30,29)(26,32,33,38)(27,35,37,36)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=ehi,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 193444);