#242870

order	512
nilpotency class	3
isoclinism	isoclinic class of #233019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

696 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)(9,10,14,22,13,17,12,16)(11,18,19,15,21,23,20,24)(25,26)(27,30)(28,32)(29,31)(33,37)(34,39)(35,38)(36,40);(2,5)(6,8)(25,27,29,35)(26,30,31,38)(28,33,36,34)(32,37,40,39);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,24)(18,22)(27,33)(28,36)(30,37)(32,40)(34,35)(38,39);(9,12,13,14)(10,16,17,22)(11,19,21,20)(15,23,24,18)(26,31)(30,38)(32,40)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150066);