#461757

order	512
nilpotency class	3
isoclinism	isoclinic class of #455116
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368953);