#329380

order	512
nilpotency class	3
isoclinism	isoclinic class of #327021
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

714 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ei,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi,d^c=de,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, 236576);