#286452

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
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

581 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)(9,10,15,20,14,19,13,18)(11,21,22,24,12,16,23,17)(25,26,31,33,30,28,29,34)(27,32,35,40,37,38,36,39);(2,5)(6,8)(9,11,13,23,14,12,15,22)(10,16,18,24,19,21,20,17)(25,27,31,35,30,37,29,36)(26,32,33,40,28,38,34,39);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,17,19,24)(13,23,15,22)(16,18,21,20)(25,28,30,26)(27,32,37,38)(29,34,31,33)(35,39,36,40);(2,5)(6,8)(9,13,14,15)(10,18,19,20)(11,22,12,23)(16,17,21,24)(25,29,30,31)(26,33,28,34)(27,35,37,36)(32,39,38,40)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=ei,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,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, 193648);