#7193173

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	#3512	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1554 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)(19,31)(20,36)(21,27)(22,28)(23,29)(24,30)(25,34)(26,32)(33,44)(35,47)(37,45)(38,43)(39,46)(40,48)(41,42);(1,3,5,10)(2,6,9,7)(4,13,14,11)(8,16,12,15)(17,19)(18,25)(20,21)(22,33)(23,34)(24,35)(26,42)(27,45)(28,31)(29,47)(30,44)(32,40)(36,48)(37,39)(38,41)(43,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,16)(12,15)(13,14)(17,20,22,37,24,38,23,32)(18,26,29,43,30,45,28,36)(19,21,33,39,35,41,34,40)(25,42,47,46,44,27,31,48);(3,10)(6,7)(11,13)(15,16)(17,21)(18,27)(19,32)(20,33)(22,39)(23,40)(24,41)(25,43)(26,31)(28,46)(29,48)(30,42)(34,38)(35,37)(36,44)(45,47);(1,4,5,14)(2,8,9,12)(3,11,10,13)(6,15,7,16)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,37,38,32)(21,39,41,40)(25,31,44,47)(26,36,45,43)(27,46,42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,e^2=hi,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, 7100369);