#205531

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
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	#60782	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

998 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:

(1,2,5,10,6,11,4,9)(3,8,12,15,13,16,14,7)(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)(2,7)(4,12)(5,14)(6,13)(8,9)(10,16)(11,15)(17,19)(18,24,27,39)(20,29)(21,31)(22,32)(23,30)(25,38,40,43)(26,28,37,42)(33,41,47,48)(34,46)(35,44)(36,45);(1,3)(2,8)(4,14)(5,12)(6,13)(7,9)(10,15)(11,16)(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);(1,4,6,5)(2,9,11,10)(3,12,13,14)(7,8,15,16)(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=egi,c^2=f,d^2=i,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=gi,d^c=dhi

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, 112727);