#313893

order	512
nilpotency class	3
isoclinism	isoclinic class of #264203
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

647 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)(11,21,13,17)(12,16,23,18)(14,24,22,19)(25,26)(27,34)(28,38)(29,32)(30,33)(31,37)(35,39)(36,40);(2,5)(6,8)(9,11,14,12,15,13,22,23)(10,16,19,21,20,18,24,17)(25,27)(26,31)(28,36)(29,37)(30,35)(32,34)(33,40)(38,39);(1,3,4,7)(2,6,5,8)(9,12,22,11,15,23,14,13)(10,17,24,18,20,21,19,16)(26,32)(27,35)(28,30)(31,39)(34,40)(36,37);(2,5)(6,8)(9,13,14,23,15,11,22,12)(10,18,19,17,20,16,24,21)(25,28,29,30)(26,33,32,38)(27,36,37,35)(31,40,34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=ehi,d^2=f,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,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 221089);