#6367450

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	#3482	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1482 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,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, 6274646);