#117304

order	512
nilpotency class	3
isoclinism	isoclinic class of #117289
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=hi,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,f^b=fhi,d^c=dh,e^c=eh,f^c=fi,g^c=ghi

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, 24500);