#319745

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
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

760 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)(11,14)(12,15)(13,16)(17,24)(18,22)(19,21)(20,23)(25,26,30,34,31,28,29,33)(27,32,38,35,36,39,37,40);(1,3,4,8)(2,5,6,7)(9,11,13,19,12,18,20,17)(10,14,16,21,15,22,23,24)(25,27,30,38,31,36,29,37)(26,32,34,35,28,39,33,40);(10,15)(11,17)(13,20)(14,21)(18,19)(22,24)(25,28,31,26)(27,35,36,40)(29,33,30,34)(32,37,39,38);(2,6)(5,7)(27,36)(32,39)(35,40)(37,38)

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,c^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226941);