#6356787

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356733
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

1488 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)(4,12)(7,13)(8,14)(9,16)(11,15)(17,18,23,28)(19,29,31,40)(20,34,36,42)(21,27,22,26)(24,33,39,32)(25,37,41,35)(30,44,45,38)(43,47,48,46);(1,2,4,6,5,8,12,14)(3,9,11,16,10,7,15,13)(17,19,21,32,23,31,22,33)(18,24,26,29,28,39,27,40)(20,30,35,46,36,45,37,47)(25,38,34,43,41,44,42,48);(17,20,23,36)(18,25,28,41)(19,30,31,45)(21,35,22,37)(24,38,39,44)(26,34,27,42)(29,43,40,48)(32,46,33,47);(1,3)(2,7)(4,11)(5,10)(6,13)(8,9)(12,15)(14,16)(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47);(3,10)(7,9)(11,15)(13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263983);