#307387

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
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	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

632 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)(4,5)(6,8)(9,10)(11,17)(12,16)(13,20)(14,19)(15,21)(18,23)(22,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(2,5)(6,7)(9,11,14,22,15,23,13,12)(10,16,20,18,21,24,19,17)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,37)(39,40);(9,12,13,23,15,22,14,11)(10,17,19,24,21,18,20,16)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(1,3,4,8)(2,6,5,7)(9,12,13,23,15,22,14,11)(10,18,19,16,21,17,20,24)(26,31)(30,33)(32,40)(38,39)

Polycyclic presentation:

2^9|b^2=f,c^2=fh,d^2=ei,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,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 214583);