#286022

order	512
nilpotency class	3
isoclinism	isoclinic class of #281915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

563 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)(9,10,14,19,15,20,13,18)(11,16,21,17,12,22,23,24)(25,26,29,32,28,31,36,37)(27,33,35,40,34,30,38,39);(2,5)(6,8)(9,11,14,21,15,12,13,23)(10,16,19,17,20,22,18,24)(25,27,29,35,28,34,36,38)(26,30,32,39,31,33,37,40);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,15)(13,23)(14,21)(16,19)(18,22)(20,24)(25,26)(27,30)(28,31)(29,37)(32,36)(33,34)(35,40)(38,39);(2,5)(6,8)(9,13,15,14)(10,18,20,19)(11,21,12,23)(16,17,22,24)(26,31)(27,34)(32,37)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 193218);