#313243

order	512
nilpotency class	3
isoclinism	isoclinic class of #264427
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

597 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)(9,10,14,18)(11,19,20,16)(12,15,21,24)(13,22,23,17)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,3,4,8)(2,5,6,7)(9,11,13,12,14,20,23,21)(10,15,17,19,18,24,22,16)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,37)(39,40);(2,6)(5,7)(11,20)(12,21)(15,24)(16,19)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(2,6)(5,7)(9,12,13,20,14,21,23,11)(10,16,17,15,18,19,22,24)(26,31)(30,33)(32,40)(38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,d^2=efi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=gi,d^c=dh,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, 220439);