#307686

order	512
nilpotency class	3
isoclinism	isoclinic class of #264363
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×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

592 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)(9,10,16,21)(11,18,22,24)(12,17,13,23)(14,20,15,19)(25,26)(27,33)(28,31)(29,32)(30,35)(34,38)(36,40)(37,39);(2,5)(6,7)(9,11,15,13,16,22,14,12)(10,17,20,24,21,23,19,18)(25,27)(26,30)(28,35)(29,34)(31,33)(32,40)(36,37)(38,39);(9,12,15,11,16,13,14,22)(10,18,20,17,21,24,19,23)(26,31)(27,34)(29,37)(30,38)(33,40)(35,36);(1,3,4,8)(2,6,5,7)(9,13,15,22,16,12,14,11)(10,18,20,17,21,24,19,23)(26,31)(30,33)(32,39)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 214882);