#6354898

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354835
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	#60917	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1556 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262094);