#250389

order	512
nilpotency class	3
isoclinism	isoclinic class of #250327
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	#62238	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

956 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157585);