#6354518

order	512
nilpotency class	3
isoclinism	isoclinic class of #19420
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#121
prime radical	#60559	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1611 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)(3,10,11,16)(4,8,5,14)(7,12,15,13)(17,18,23,28)(19,29,31,41)(20,34,37,44)(21,27,22,26)(24,33,40,32)(25,38,43,36)(30,42,35,39)(45,47,48,46);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(17,19,21,32,23,31,22,33)(18,24,26,29,28,40,27,41)(20,35,36,47,37,30,38,46)(25,42,34,48,43,39,44,45);(3,11)(7,15)(10,16)(12,13)(17,20)(18,25)(19,30)(21,36)(22,38)(23,37)(24,39)(26,34)(27,44)(28,43)(29,45)(31,35)(32,46)(33,47)(40,42)(41,48);(19,31)(24,40)(29,41)(30,35)(32,33)(39,42)(45,48)(46,47);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)

Polycyclic presentation:

2^9|a^2=h,b^2=f,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=di,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, 6261714);