#6356816

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

1493 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)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,35)(30,44)(31,40)(33,39)(36,41)(37,42)(38,45)(43,46)(47,48);(1,3,5,14,7,15,6,9)(2,8,13,16,11,4,10,12)(17,19,22,33,23,31,21,32)(18,24,26,40,28,39,27,29)(20,30,37,47,36,45,35,46)(25,38,41,48,42,44,34,43);(17,20,23,36)(18,25,28,42)(19,30,31,45)(21,35,22,37)(24,38,39,44)(26,41,27,34)(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)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,36,37)(24,40,39,29)(25,41,42,34)(30,46,45,47)(38,48,44,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=i,e^2=h,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, 6264012);