#263586

order	512
nilpotency class	3
isoclinism	isoclinic class of #263555
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19053	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

808 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)(3,7)(4,9)(5,8)(6,10)(11,14)(12,16)(13,15)(17,18,23,29)(19,21,33,42)(20,34,39,45)(22,28,30,27)(24,32,41,26)(25,37,46,38)(31,43,47,48)(35,36,44,40);(1,3,6,12)(2,7,10,16)(4,11,5,13)(8,15,9,14)(17,19,30,32,23,33,22,26)(18,24,28,42,29,41,27,21)(20,35,38,31,39,44,37,47)(25,36,34,48,46,40,45,43);(2,8)(4,5)(7,14)(9,10)(11,13)(15,16)(17,20)(18,25)(19,31)(21,40)(22,37)(23,39)(24,43)(26,35)(27,45)(28,34)(29,46)(30,38)(32,44)(33,47)(36,42)(41,48);(1,4,6,5)(2,9,10,8)(3,11,12,13)(7,14,16,15)(17,21,22,41,23,42,30,24)(18,26,27,33,29,32,28,19)(20,36,37,43,39,40,38,48)(25,44,45,31,46,35,34,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,g^b=gh,d^c=di,f^c=fi,g^c=gh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170782);