#205139

order	512
nilpotency class	3
isoclinism	isoclinic class of #19304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,22,26,23,27,21,37)(19,28,32,39,30,42,31,24)(20,33,36,40,34,47,35,25)(29,43,45,48,46,38,44,41);(1,3,4,12,5,11,13,10)(2,6,8,16,7,14,15,9)(17,19,23,30)(18,24)(20,29,34,46)(21,31,22,32)(25,38)(26,28)(27,39)(33,41)(35,44,36,45)(37,42)(40,43)(47,48);(17,20,21,35,23,34,22,36)(18,25,26,33,27,40,37,47)(19,29,31,44,30,46,32,45)(24,38,28,41,39,43,42,48);(2,7)(6,14)(8,15)(9,16)(19,30)(20,34)(24,39)(25,40)(28,42)(31,32)(33,47)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112335);