#245870

order	512
nilpotency class	3
isoclinism	isoclinic class of #245807
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	#61717	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1108 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)(19,30)(20,34)(21,37)(22,28)(23,27)(24,29)(25,32)(26,36)(31,40)(33,44)(35,43)(38,42)(39,41)(45,46)(47,48);(2,7)(4,5)(8,14)(9,10)(11,12)(15,16)(17,19,24,33)(18,25,29,41)(20,31,21,38)(22,39,23,32)(26,42,43,40)(27,44,28,30)(34,45,37,48)(35,47,36,46);(1,3,6,13)(2,8,10,16)(4,11,5,12)(7,14,9,15)(17,20,24,21)(18,26,29,43)(19,31,33,38)(22,36,23,35)(25,40,41,42)(27,34,28,37)(30,45,44,48)(32,47,39,46);(1,4)(3,11)(5,6)(7,9)(12,13)(14,15)(17,21,24,20)(18,26,29,43)(19,31,33,38)(22,35,23,36)(25,40,41,42)(27,34,28,37)(30,45,44,48)(32,47,39,46)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,e^c=ei,f^c=fi,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, 153066);