#6356896

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	#61998	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1504 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:

(1,2,7,11)(3,12,15,16)(4,9,8,14)(5,10,6,13)(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);(1,3,5,14,7,15,6,9)(2,8,13,16,11,4,10,12)(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);(1,4)(2,9)(3,13)(5,12)(6,16)(7,8)(10,15)(11,14)(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47);(1,5,7,6)(2,10,11,13)(3,14,15,9)(4,16,8,12)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=i,e^2=hi,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, 6264092);