#238162

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

870 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)(11,14)(12,20)(13,17)(15,19)(16,22)(18,23)(21,24)(28,36)(30,39)(32,37)(34,38);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,19)(13,18)(15,20)(16,24)(17,23)(21,22)(25,26)(27,33)(28,37)(29,31)(30,34)(32,36)(35,40)(38,39);(9,11)(10,15)(12,21)(13,18)(14,20)(16,23)(17,24)(19,22)(25,27,29,35)(26,30,31,39)(28,38,36,34)(32,40,37,33);(11,18)(14,23)(15,24)(19,21)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^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, 145358);