#205573

order	512
nilpotency class	3
isoclinism	isoclinic class of #205569
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

960 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)(3,10)(4,9)(5,14)(6,8)(7,11)(12,16)(13,15)(17,18)(19,28,31,39)(20,32)(21,26)(22,37)(23,27)(24,30,40,36)(25,35)(29,38,43,48)(33,46)(34,41)(42,47,45,44);(1,3,5,11,6,12,4,13)(2,7,9,10,8,15,14,16)(17,19,21,36,23,31,22,30)(18,24,26,39,27,40,37,28)(20,29,34,47,33,43,35,44)(25,38,32,45,41,48,46,42);(2,8)(7,15)(9,14)(10,16)(17,20,22,35,23,33,21,34)(18,25,26,32,27,41,37,46)(19,29,30,44,31,43,36,47)(24,38,39,45,40,48,28,42);(1,4,6,5)(2,9,8,14)(3,11,12,13)(7,16,15,10)(20,33)(25,41)(29,43)(32,46)(34,35)(38,48)(42,45)(44,47)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=h,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112769);