#116116

order	512
nilpotency class	3
isoclinism	isoclinic class of #10382
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

695 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)(3,5)(4,7)(6,8)(9,10,14,20,15,19,13,18)(11,16,24,32,23,33,22,31)(12,25,29,17,27,35,28,34)(21,36,38,30,26,39,37,40);(3,8)(5,6)(9,11)(10,16)(12,26,27,21)(13,22)(14,24)(15,23)(17,30,34,40)(18,31)(19,33)(20,32)(25,36,35,39)(28,38,29,37);(2,5)(3,8)(6,7)(9,12)(10,17)(11,21)(13,29)(14,28)(15,27)(16,30)(18,35)(19,34)(20,25)(22,38)(23,26)(24,37)(31,39)(32,36)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,g^2=h,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,e^c=ehi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23312);