#6367521

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367198
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#61985	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1503 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh

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, 6274717);