#250620

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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1046 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)(19,24,31,40)(20,32,34,43)(21,27,22,26)(25,36,42,35)(29,41,45,48)(30,44,37,39)(33,38,46,47);(2,7)(3,5)(8,10)(9,12)(11,14)(15,16)(17,19)(18,24)(20,33)(21,30)(22,37)(23,31)(25,41)(26,39)(27,44)(28,40)(29,36)(32,47)(34,46)(35,45)(38,43)(42,48);(1,3,6,5)(2,8,10,7)(4,11,13,14)(9,15,16,12)(17,20)(18,25)(19,29)(21,36)(22,35)(23,34)(24,38)(26,43)(27,32)(28,42)(30,46)(31,45)(33,37)(39,48)(40,47)(41,44);(1,4,6,13)(2,9,10,16)(3,11,5,14)(7,12,8,15)(17,19)(18,24)(20,29)(21,37)(22,30)(23,31)(25,38)(26,44)(27,39)(28,40)(32,48)(33,36)(34,45)(35,46)(41,43)(42,47)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=h,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, 157816);