#250070

order	512
nilpotency class	3
isoclinism	isoclinic class of #249759
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	#61868	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

947 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,g^c=gh,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, 157266);