#205133

order	512
nilpotency class	3
isoclinism	isoclinic class of #19328
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	C₂⁴	#124
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)(20,33,35,40,36,47,34,25)(29,43,46,48,44,38,45,41);(1,3,4,12,5,11,13,10)(2,6,8,16,7,14,15,9)(17,19,23,30)(18,24)(20,29,36,44)(21,31,22,32)(25,38)(26,28)(27,39)(33,41)(34,45,35,46)(37,42)(40,43)(47,48);(17,20,22,35,23,36,21,34)(18,25,37,47,27,40,26,33)(19,29,32,46,30,44,31,45)(24,38,42,48,39,43,28,41);(2,7)(6,14)(8,15)(9,16)(19,30)(24,39)(28,42)(29,44)(31,32)(38,43)(41,48)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112329);