#192587

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,d^b=dh,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fh

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, 99783);