#122085

order	512
nilpotency class	3
isoclinism	isoclinic class of #122083
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=eh,g^b=gi,e^c=eh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29281);