#6369191

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369150
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1515 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)(8,15)(9,16)(10,14)(11,13)(17,18,22,25)(19,23,28,39)(20,30,33,42)(21,34,36,26)(24,35,41,32)(27,43,31,48)(29,46,45,38)(37,47,40,44);(1,2)(3,8)(4,7)(5,14)(6,10)(9,11)(12,16)(13,15)(17,19)(18,23)(20,31)(21,29)(22,28)(24,40)(25,39)(26,38)(27,33)(30,48)(32,47)(34,46)(35,44)(36,45)(37,41)(42,43);(1,3)(2,8)(4,13)(5,11)(6,12)(7,15)(9,14)(10,16)(17,20,22,33)(18,24,25,41)(19,27,28,31)(21,35,36,32)(23,37,39,40)(26,42,34,30)(29,47,45,44)(38,48,46,43);(1,3)(2,9)(4,11)(5,13)(6,12)(7,16)(8,14)(10,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,37)(25,39)(26,38)(30,43)(31,33)(32,44)(34,46)(35,47)(36,45)(40,41)(42,48);(1,4,6,5)(2,7,10,14)(3,11,12,13)(8,9,16,15)(18,25)(19,28)(24,41)(26,34)(27,31)(29,45)(30,42)(44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276387);