#253673

order	512
nilpotency class	3
isoclinism	isoclinic class of #253651
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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

698 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)(3,4)(5,9)(6,10)(7,11)(8,14)(12,16)(13,15)(17,18,22,27)(19,23,29,25)(20,24,32,40)(21,26,35,44)(28,39,45,36)(30,37,46,42)(31,48,47,38)(33,41,34,43);(1,3,7,13)(2,4,11,15)(5,8,6,12)(9,14,10,16)(17,19)(18,23)(20,31,32,47)(21,30)(22,29)(24,38,40,48)(25,27)(26,37)(28,33,45,34)(35,46)(36,41,39,43)(42,44);(1,4)(2,8)(3,10)(5,14)(6,16)(7,15)(9,13)(11,12)(17,20)(18,24)(19,28)(21,34)(22,32)(23,36)(25,39)(26,43)(27,40)(29,45)(30,47)(31,46)(33,35)(37,48)(38,42)(41,44);(1,5,7,6)(2,9,11,10)(3,8,13,12)(4,14,15,16)(17,19)(18,25)(20,28)(21,30)(22,29)(23,27)(24,39)(26,42)(31,33)(32,45)(34,47)(35,46)(36,40)(37,44)(38,43)(41,48)

Polycyclic presentation:

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

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, 160869);