#6355594

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355552
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1500 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,e^b=eh,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, 6262790);