#7183660

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996395
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×AGL(1,ℤ/8)	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=hi,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi

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, 7090856);