#6354244

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354237
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

1558 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,23,27)(19,28,31,42)(20,25,34,40)(21,26,22,36)(24,30,38,32)(29,41,45,48)(33,39,35,46)(37,43,47,44);(17,19,22,32,23,31,21,30)(18,24,26,28,27,38,36,42)(20,29,35,44,34,45,33,43)(25,37,39,41,40,47,46,48);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,37)(26,39)(27,40)(28,41)(30,43)(31,45)(32,44)(36,46)(38,47)(42,48);(2,7)(6,15)(8,12)(9,16)(17,21,23,22)(18,26,27,36)(19,30,31,32)(20,33,34,35)(24,28,38,42)(25,39,40,46)(29,43,45,44)(37,41,47,48);(19,31)(20,34)(24,38)(25,40)(28,42)(30,32)(33,35)(39,46)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,e^b=eh,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, 6261440);