#250866

order	512
nilpotency class	3
isoclinism	isoclinic class of #250775
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	#447	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1011 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158062);