#315955

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

640 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)(11,17)(12,21)(13,15)(14,19)(16,18)(20,23)(22,24)(27,33)(28,35)(30,37)(32,38)(34,39)(36,40);(1,3)(2,5)(4,7)(6,8)(9,11,18,24,13,20,12,19)(10,14,21,23,15,22,16,17)(25,26)(27,34)(28,32)(29,31)(30,36)(33,39)(35,38)(37,40);(3,7)(5,8)(10,15)(14,22)(16,21)(17,23)(25,27,35,40,29,36,28,33)(26,30,38,39,31,34,32,37);(9,11,18,24,13,20,12,19)(10,14,21,23,15,22,16,17)(26,31)(30,34)(32,38)(37,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=eh,c^2=fhi,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di

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, 223151);