#7193166

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	#61278	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1588 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:

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

Polycyclic presentation:

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

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, 7100362);