#250570

order	512
nilpotency class	3
isoclinism	isoclinic class of #250519
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

975 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157766);