#466974

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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 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)(11,21)(12,26)(13,16)(14,19)(15,20)(17,24)(18,30)(22,37)(23,34)(25,31)(27,40)(28,36)(29,32)(33,38)(35,39);(2,5)(6,8)(10,16,20,19)(17,21,34,31)(18,32,26,36)(33,37,40,35);(1,3,4,7)(2,6,5,8)(9,11)(10,17)(12,27)(13,24)(14,23)(15,25)(16,31)(18,35)(19,21)(20,34)(22,28)(26,37)(29,39)(30,38)(32,40)(33,36);(3,7)(6,8)(9,12,13,28,15,30,14,29)(10,18,19,36,20,26,16,32)(11,22,23,27,25,39,24,38)(17,33,31,35,34,40,21,37)

Polycyclic presentation:

2^9|a^2=ef,b^2=g,c^2=ef,d^2=fi,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, 374170);