#205521

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
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

984 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:

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

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, 112717);