#507547

order	512
nilpotency class	4
isoclinism	isoclinic class of #500891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

519 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)(3,7)(4,6)(5,8)(9,10)(11,22)(12,16)(13,21)(14,19)(15,30)(17,27)(18,23)(20,28)(24,33)(25,37)(26,35)(29,32)(31,38)(34,40)(36,39);(1,3,4,8)(2,5,6,7)(9,11,23,24,14,12,13,26)(10,16,18,35,19,22,21,33)(15,25,39,40,29,27,28,31)(17,36,38,32,37,20,34,30);(9,12,14,11)(10,17)(13,26,23,24)(15,31,29,40)(16,32)(18,34)(19,37)(20,33)(21,38)(22,30)(25,39,27,28)(35,36);(2,6)(5,7)(9,13,14,23)(10,18,19,21)(11,24,12,26)(15,28,29,39)(16,33,22,35)(17,34,37,38)(20,30,36,32)(25,40,27,31)

Polycyclic presentation:

2^9|b^2=eg,c^2=gh,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307037, #72307041, #72307046
maximal normal subgroup of	#72307037, #72307041, #72307046

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414743);