#250154

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
prime radical	#3035	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1001 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)(17,18,24,20)(19,29,34,30)(21,37,39,42)(22,28,23,27)(25,33,35,32)(26,40,36,38)(31,41,46,48)(43,47,44,45);(2,7)(3,5)(8,10)(9,12)(11,14)(15,16)(17,19,24,34)(18,25,20,35)(21,31,39,46)(22,33,23,32)(26,41,36,48)(27,30,28,29)(37,43,42,44)(38,47,40,45);(1,3,6,5)(2,8,10,7)(4,11,13,14)(9,15,16,12)(17,20,24,18)(19,30,34,29)(21,36,39,26)(22,27,23,28)(25,32,35,33)(31,44,46,43)(37,40,42,38)(41,47,48,45);(1,4,6,13)(2,9,10,16)(3,11,5,14)(7,12,8,15)(17,21,24,39)(18,26,20,36)(19,31,34,46)(22,40,23,38)(25,41,35,48)(27,42,28,37)(29,43,30,44)(32,47,33,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,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, 157350);