#7183667

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	#2002	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1577 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,d^2=fi,e^2=hi,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, 7090863);