#205554

order	512
nilpotency class	3
isoclinism	isoclinic class of #205553
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60782	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

991 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,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, 112750);