#7183678

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	#3041	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1570 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)(18,24,28,27)(19,29,32,31)(20,33)(25,38)(26,40,41,34)(30,44,46,43)(35,36)(37,42);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,18)(19,25)(20,34)(21,24)(22,27)(23,28)(26,36)(29,37)(30,45)(31,42)(32,38)(33,40)(35,41)(39,44)(43,47)(46,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,19)(18,25)(20,30)(21,29)(22,31)(23,32)(24,37)(26,39)(27,42)(28,38)(33,46)(34,45)(35,43)(36,44)(40,48)(41,47);(3,10)(6,9)(11,16)(14,15)(17,20,22,36,23,33,21,35)(18,26,24,34,28,41,27,40)(19,30,31,44,32,46,29,43)(25,39,37,45,38,47,42,48);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(17,21,23,22)(18,24,28,27)(19,29,32,31)(20,35,33,36)(25,37,38,42)(26,34,41,40)(30,43,46,44)(39,45,47,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,d^2=f,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, 7090874);