#250281

order	512
nilpotency class	3
isoclinism	isoclinic class of #250199
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	#3499	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1037 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)(4,9)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,24,20)(19,29,34,30)(21,37,40,44)(22,28,23,27)(25,33,35,32)(26,41,36,39)(31,42,38,43)(45,48,46,47);(2,7)(4,5)(8,14)(9,10)(11,12)(15,16)(17,19,24,34)(18,25,20,35)(21,38,40,31)(22,33,23,32)(26,43,36,42)(27,30,28,29)(37,46,44,45)(39,47,41,48);(1,3,6,13)(2,8,10,16)(4,11,5,12)(7,14,9,15)(17,20,24,18)(19,30,34,29)(21,36,40,26)(22,27,23,28)(25,32,35,33)(31,46,38,45)(37,41,44,39)(42,48,43,47);(1,4)(3,11)(5,6)(7,9)(12,13)(14,15)(17,21)(18,26)(19,31)(20,36)(22,41)(23,39)(24,40)(25,42)(27,44)(28,37)(29,45)(30,46)(32,48)(33,47)(34,38)(35,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^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,d^b=dh,e^b=eh,g^b=gi,e^c=ei,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157477);