#252984

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

883 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)(11,15)(12,17)(13,22)(14,18)(16,19)(20,24)(21,23)(25,26,30,34)(27,35,38,39)(28,32,29,33)(31,36,40,37);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,21)(13,19)(14,20)(16,22)(17,23)(18,24)(27,36)(28,29)(31,39)(32,33)(35,40)(37,38);(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,24)(18,23)(25,27,30,38)(26,31,34,40)(28,36,29,37)(32,35,33,39);(2,6)(5,7)(9,12,14,13)(10,17,18,22)(11,19,20,21)(15,16,24,23)(25,28,30,29)(26,32,34,33)(27,36,38,37)(31,35,40,39)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=h,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, 160180);