#6356570

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=h,e^2=h,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, 6263766);