#250126

order	512
nilpotency class	3
isoclinism	isoclinic class of #250071
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1040 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157322);