#6346928

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
prime radical	#62781	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2815 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)(5,13)(6,9)(7,12)(11,16)(14,15)(17,18,22,26)(19,23,29,36)(20,24,31,38)(21,32,33,25)(27,39,41,34)(28,42,43,35)(30,44,45,37)(40,46,48,47);(1,3)(2,7)(4,12)(5,14)(6,11)(8,16)(9,15)(10,13)(17,19)(18,23)(20,27)(21,28)(22,29)(24,34)(25,35)(26,36)(30,40)(31,41)(32,42)(33,43)(37,46)(38,39)(44,47)(45,48);(17,20)(18,24)(19,27)(21,30)(22,31)(23,34)(25,37)(26,38)(28,40)(29,41)(32,44)(33,45)(35,46)(36,39)(42,47)(43,48);(3,11)(7,15)(10,16)(12,14)(17,21,22,33)(18,25,26,32)(19,28,29,43)(20,30,31,45)(23,35,36,42)(24,37,38,44)(27,40,41,48)(34,46,39,47);(1,4,6,5)(2,8,9,13)(3,12,11,14)(7,16,15,10)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254124);