#250628

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

1070 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,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, 157824);