#6359939

order	512
nilpotency class	3
isoclinism	isoclinic class of #6359920
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	C₂×#550	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1585 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=chi,d^b=df,f^b=fh,g^c=gi,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, 6267135);