#314596

order	512
nilpotency class	3
isoclinism	isoclinic class of #264555
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

634 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)(3,6)(4,5)(7,8)(9,10,15,20)(11,21,23,17)(12,24,22,16)(13,19,14,18)(25,26,29,33)(27,34,37,31)(28,38,30,32)(35,40,36,39);(2,5)(6,8)(9,11,14,12,15,23,13,22)(10,16,19,17,20,24,18,21)(25,27,29,37)(26,31,33,34)(28,36,30,35)(32,39,38,40);(1,3,4,7)(2,6,5,8)(9,12,13,11,15,22,14,23)(10,17,18,16,20,21,19,24)(27,35)(28,30)(31,39)(32,38)(34,40)(36,37);(2,5)(6,8)(9,11,13,22,15,23,14,12)(10,16,18,21,20,24,19,17)(25,28,29,30)(26,32,33,38)(27,36,37,35)(31,39,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 221792);