#6354446

order	512
nilpotency class	3
isoclinism	isoclinic class of #19384
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1653 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,22,26)(19,23,28,37)(20,30,32,40)(21,33,35,25)(24,34,39,31)(27,41,42,48)(29,45,44,38)(36,46,47,43);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,36)(25,38)(26,37)(30,41)(31,43)(32,42)(33,45)(34,46)(35,44)(39,47)(40,48);(17,20,22,32)(18,24,26,39)(19,27,28,42)(21,34,35,31)(23,36,37,47)(25,40,33,30)(29,46,44,43)(38,48,45,41);(19,28)(23,37)(27,42)(29,44)(36,47)(38,45)(41,48)(43,46);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261642);