#265218

order	512
nilpotency class	3
isoclinism	isoclinic class of #265163
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	C₂³	#546
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

771 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,15)(14,16)(17,18,23,28)(19,29,33,40)(20,34,37,44)(21,27,22,26)(24,32,41,31)(25,38,43,36)(30,42,35,39)(45,47,48,46);(1,2)(3,12)(4,8)(5,9)(6,10)(7,14)(11,15)(13,16)(17,19,22,32,23,33,21,31)(18,24,27,40,28,41,26,29)(20,30,38,47,37,35,36,46)(25,39,44,48,43,42,34,45);(1,3,6,15)(2,7,10,16)(4,14,5,13)(8,11,9,12)(17,20,23,37)(18,25,28,43)(19,30,33,35)(21,36,22,38)(24,39,41,42)(26,34,27,44)(29,45,40,48)(31,46,32,47);(1,4,6,5)(2,8,10,9)(3,13,15,14)(7,12,16,11)(17,19,21,31,23,33,22,32)(18,24,26,29,28,41,27,40)(20,35,36,47,37,30,38,46)(25,42,34,48,43,39,44,45)

Polycyclic presentation:

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

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, 172414);