#265052

order	512
nilpotency class	3
isoclinism	isoclinic class of #265019
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	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

788 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)(3,11,13,15)(4,9,5,8)(7,12,16,14)(17,18)(19,27)(20,34)(21,38)(22,29)(23,28)(24,30)(25,32)(26,36)(31,47)(33,42)(35,45)(37,46)(39,43)(40,44)(41,48);(3,12)(4,5)(7,15)(8,9)(11,16)(13,14)(17,19,22,32,24,33,23,21)(18,25,28,27,30,38,29,42)(20,31,36,35,37,48,40,39)(26,41,34,45,44,47,46,43);(1,3,6,13)(2,7,10,16)(4,12,5,14)(8,15,9,11)(17,20,24,37)(18,26,30,44)(19,31,33,48)(21,39,32,35)(22,36,23,40)(25,41,38,47)(27,45,42,43)(28,34,29,46);(3,13)(7,16)(11,15)(12,14)(17,21,23,33,24,32,22,19)(18,27,29,25,30,42,28,38)(20,35,40,31,37,39,36,48)(26,43,46,47,44,45,34,41)

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=i,d^2=eh,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,f^b=fhi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

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, 172248);