#466975

order	512
nilpotency class	3
isoclinism	isoclinic class of #457090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=gi,c^2=ef,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,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, 374171);