#7183679

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	#26648	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1595 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)(20,32)(26,41,42,33)(29,44,46,43)(34,35)(38,47)(45,48);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,18)(19,25)(20,33)(21,24)(22,36)(23,27)(26,35)(28,39)(29,45)(30,37)(31,40)(32,41)(34,42)(38,43)(44,47)(46,48);(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)(32,46)(33,45)(34,44)(35,43)(36,39)(41,48)(42,47);(3,10)(6,9)(11,16)(14,15)(17,20,21,34,23,32,22,35)(18,26,36,41,27,42,24,33)(19,29,30,44,31,46,28,43)(25,38,39,48,40,47,37,45);(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)(20,34,32,35)(25,37,40,39)(26,33,42,41)(29,44,46,43)(38,45,47,48)

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,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, 7090875);