#228326

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

591 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=deh,d^c=dhi,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 135522);