#285893

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
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

609 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)(9,10,12,16,13,17,21,15)(11,14,20,23,19,22,18,24)(25,26,30,32,29,33,28,37)(27,34,35,39,36,31,38,40);(2,5)(6,8)(9,11,12,20,13,19,21,18)(10,14,16,23,17,22,15,24)(25,27,30,35,29,36,28,38)(26,31,32,40,33,34,37,39);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(12,21)(14,22)(16,17)(19,20)(25,26)(27,31)(28,32)(29,33)(30,37)(34,36)(35,39)(38,40);(2,5)(6,8)(11,19)(14,22)(18,20)(23,24)(25,28,29,30)(26,32,33,37)(27,35,36,38)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=ei,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,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, 193089);