#238166

order	512
nilpotency class	3
isoclinism	isoclinic class of #234028
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

828 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145362);