#457133

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1175 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364329);