#6566134

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535687
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	C₂²×C₄	C₂⁵
prime radical	#3475	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1621 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=f,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,g^c=gi,e^d=eh

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, 6473330);