#457136

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=i,d^2=e,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, 364332);