#7183671

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996405
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×#41	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1548 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)(19,31,34,46)(20,35,37,47)(21,38)(22,28)(23,29)(24,30)(25,33,45,41)(26,36,44,42)(27,40)(32,43)(39,48);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,24,34)(18,25,30,45)(20,32,37,39)(21,36,40,42)(22,41,23,33)(26,43,44,48)(27,47,38,35)(28,46,29,31);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20)(18,26)(19,32)(21,33)(22,36)(23,42)(24,37)(25,43)(27,46)(28,35)(29,47)(30,44)(31,38)(34,39)(40,41)(45,48);(3,10)(6,9)(11,16)(14,15)(17,21,22,39,24,40,23,32)(18,27,28,43,30,38,29,48)(19,20,33,36,34,37,41,42)(25,44,31,47,45,26,46,35);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7090867);