#6354980

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354913
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

1524 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)(3,12,14,15)(4,13,8,9)(5,10,6,16)(17,18,23,28)(19,29,33,39)(20,34,35,42)(21,27,22,26)(24,32,40,31)(25,37,41,36)(30,44,45,38)(43,47,48,46);(1,3,5,9,7,14,6,13)(2,8,16,15,11,4,10,12)(17,19,22,32,23,33,21,31)(18,24,27,39,28,40,26,29)(20,30,37,47,35,45,36,46)(25,38,42,48,41,44,34,43);(1,4)(2,9)(3,10)(5,12)(6,15)(7,8)(11,13)(14,16)(17,20)(18,25)(19,30)(21,36)(22,37)(23,35)(24,38)(26,34)(27,42)(28,41)(29,43)(31,46)(32,47)(33,45)(39,48)(40,44);(1,5,7,6)(2,10,11,16)(3,13,14,9)(4,12,8,15);(1,5,7,6)(2,10,11,16)(3,9,14,13)(4,15,8,12)(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,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,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, 6262176);