#508777

order	512
nilpotency class	4
isoclinism	isoclinic class of #508667
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13316	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

573 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)(3,6)(4,5)(7,8)(9,10,14,20,16,22,13,18)(11,23,28,17,29,35,26,36)(12,19,15,32,31,37,30,21)(24,33,25,39,40,34,27,38);(2,5)(6,8)(9,11)(10,17)(12,25)(13,28)(14,26)(15,24)(16,29)(18,35)(19,34)(20,23)(21,38)(22,36)(27,31)(30,40)(32,39)(33,37);(3,7)(6,8)(10,18)(11,24)(12,30)(13,14)(15,31)(17,33)(19,37)(20,22)(23,39)(25,26)(27,28)(29,40)(34,36)(35,38);(1,3,4,7)(2,6,5,8)(9,12,13,30,16,31,14,15)(10,19,18,21,22,37,20,32)(11,25,28,40,29,27,26,24)(17,34,35,38,36,33,23,39)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415973);