#315609

order	512
nilpotency class	3
isoclinism	isoclinic class of #298603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

656 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=f,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222805);