#6347381

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347363
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62825	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254577);