#253356

order	512
nilpotency class	3
isoclinism	isoclinic class of #253339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

737 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

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, 160552);