#6356557

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

1528 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:

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

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,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, 6263753);