#6356669

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356633
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	#61982	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1494 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)(4,11)(7,12)(8,13)(10,15)(14,16)(17,18,23,28)(19,29,33,40)(20,34,37,43)(21,26,22,27)(24,32,41,31)(25,38,44,36)(30,42,35,39)(45,47,48,46);(1,2,4,6,5,7,11,12)(3,8,10,13,9,14,15,16)(17,19,21,31,23,33,22,32)(18,24,27,29,28,41,26,40)(20,35,36,47,37,30,38,46)(25,42,34,48,44,39,43,45);(2,7)(6,12)(8,14)(13,16)(17,20,23,37)(18,25,28,44)(19,30,33,35)(21,36,22,38)(24,39,41,42)(26,43,27,34)(29,45,40,48)(31,46,32,47);(1,3,5,9)(2,8,7,14)(4,10,11,15)(6,13,12,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,37,38)(24,40,41,29)(25,43,44,34)(30,46,35,47)(39,48,42,45);(3,9)(8,14)(10,15)(13,16)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,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, 6263865);