#7182407

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
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	#537	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1572 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)(17,18,22,28,23,29,31,27)(19,30,33,24,34,42,48,43)(20,35,37,44,32,47,36,25)(21,26,39,45,40,41,38,46);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,31,48,23,34,22,33)(18,24,28,42,29,43,27,30)(20,21,37,39,32,40,36,38)(25,41,47,45,44,26,35,46);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,23,32)(18,25,29,44)(19,21,34,40)(22,36,31,37)(24,41,43,26)(27,35,28,47)(30,46,42,45)(33,38,48,39);(3,10)(6,9)(11,16)(14,15)(17,21)(18,26)(19,32)(20,34)(22,39)(23,40)(24,25)(27,46)(28,45)(29,41)(30,47)(31,38)(33,36)(35,42)(37,48)(43,44);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7089603);