#351457

order	512
nilpotency class	3
isoclinism	isoclinic class of #348318
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

594 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)(3,5)(4,6)(7,8)(9,10,14,15)(11,18,17,13)(12,20,21,16)(19,22,23,24)(25,26)(27,34)(28,30)(29,32)(31,35)(33,37)(36,39)(38,40);(1,3,4,7)(2,5,6,8)(9,11,13,10,14,17,18,15)(12,19,22,16,21,23,24,20)(26,30)(31,34)(32,33)(38,39);(2,6)(5,8)(10,15)(11,17)(12,21)(22,24)(25,27,28,35)(26,31,30,34)(29,36,37,40)(32,38,33,39);(9,12,13,22,14,21,18,24)(10,16,17,23,15,20,11,19)(26,32)(29,37)(30,33)(31,38)(34,39)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=i,d^2=ei,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=ch,d^c=dh,g^d=ghi

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, 258653);