#205975

order	512
nilpotency class	3
isoclinism	isoclinic class of #205785
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

874 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,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, 113171);