#6448073

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	C₂×#3035	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1973 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fg,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355269);