#250526

order	512
nilpotency class	3
isoclinism	isoclinic class of #250519
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#10218	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1037 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)(3,8)(4,12)(5,7)(6,10)(9,11)(13,15)(14,16)(19,28)(21,22)(24,29)(26,36)(30,45)(31,32)(33,35)(37,47)(38,42)(39,40)(43,46)(44,48);(2,7)(3,5)(8,10)(9,12)(11,14)(15,16)(17,18)(19,29)(20,25)(21,26)(22,36)(23,27)(24,32)(28,38)(30,44)(31,42)(33,39)(34,41)(35,40)(37,45)(43,48)(46,47);(1,3,6,5)(2,8,10,7)(4,11,13,14)(9,15,16,12)(17,19)(18,24)(20,30)(21,32)(22,28)(23,31)(25,37)(26,42)(27,38)(29,36)(33,43)(34,46)(35,45)(39,44)(40,47)(41,48);(1,4,6,13)(2,9,10,16)(3,11,5,14)(7,12,8,15)(17,20,23,34)(18,25,27,41)(19,30,31,46)(21,35,22,33)(24,37,38,48)(26,40,36,39)(28,43,32,45)(29,44,42,47)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157722);