#252882

order	512
nilpotency class	3
isoclinism	isoclinic class of #247207
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

864 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,17,18,24)(12,20,22,15)(14,21,23,19)(25,26)(27,35)(28,31)(29,34)(30,33)(32,38)(36,39)(37,40);(1,3,4,8)(2,5,6,7)(26,31)(27,36)(29,30)(32,35)(37,38)(39,40);(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,20)(19,22)(25,27,28,37)(26,32,31,39)(29,36,30,38)(33,40,34,35);(2,6)(5,7)(11,18)(14,23)(17,24)(19,21)(26,31)(27,37)(33,34)(36,38)

Polycyclic presentation:

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

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, 160078);