#6355619

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355609
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60951	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1602 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,13)(15,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,42)(30,38)(32,43)(33,39)(34,40)(35,41)(37,45)(44,48)(46,47);(1,3)(2,6)(4,11)(5,10)(7,13)(8,14)(9,15)(12,16)(17,19,22,31,23,32,21,30)(18,24,26,28,27,38,36,43)(20,29,34,45,35,46,33,44)(25,37,39,42,41,48,40,47);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,25)(19,29)(21,33)(22,34)(23,35)(24,37)(26,39)(27,41)(28,42)(30,44)(31,45)(32,46)(36,40)(38,48)(43,47);(3,10)(6,14)(11,16)(13,15)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,35,34)(24,28,38,43)(25,39,41,40)(29,44,46,45)(37,42,48,47);(2,8)(6,14)(7,9)(13,15)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,34,35,33)(24,28,38,43)(25,40,41,39)(29,45,46,44)(37,47,48,42)

Polycyclic presentation:

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

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, 6262815);