#6356578

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356533
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	#3482	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1529 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263774);