#505658

order	512
nilpotency class	4
isoclinism	isoclinic class of #500599
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	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

483 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239821, #72239829, #72239840
maximal normal subgroup of	#72239821, #72239829, #72239840

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412854);