#252966

order	512
nilpotency class	3
isoclinism	isoclinic class of #247527
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

814 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)(9,10,13,15)(11,19,21,24)(12,18,14,17)(16,20,23,22)(26,31)(28,30)(34,40)(35,38);(1,3,4,8)(2,5,6,7)(10,15)(11,20)(12,14)(16,19)(21,22)(23,24)(25,26)(27,34)(28,33)(29,31)(30,37)(32,35)(36,40)(38,39);(9,11,13,21)(10,16,15,23)(12,20,14,22)(17,24,18,19)(25,27)(26,32)(28,35)(29,36)(30,38)(31,39)(33,40)(34,37);(2,6)(5,7)(9,12,13,14)(10,17,15,18)(11,20,21,22)(16,24,23,19)(25,28,29,30)(26,33,31,37)(27,35,36,38)(32,40,39,34)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=h,d^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,d^b=dh,f^b=fi,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160162);