#6482195

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470811
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	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1570 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)(9,13)(10,14)(11,15)(12,16)(17,18,22,26)(19,27,30,41)(20,32,34,24)(21,35,28,25)(23,36,38,31)(29,43,44,40)(33,39,42,47)(37,45,46,48);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19,28,36,22,30,21,31)(18,23,35,41,26,38,25,27)(20,29,42,48,34,44,33,45)(24,37,47,43,32,46,39,40);(3,9)(6,13)(10,11)(14,15)(17,20,21,33,22,34,28,42)(18,24,25,39,26,32,35,47)(19,29,31,45,30,44,36,48)(23,37,27,40,38,46,41,43);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15);(2,8)(6,13)(7,16)(14,15)(19,30)(23,38)(27,41)(29,44)(31,36)(37,46)(40,43)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,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, 6389391);