#129697

order	512
nilpotency class	3
isoclinism	isoclinic class of #8412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1779
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

529 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2)(3,11)(4,15,17,8)(5,18)(6,10)(7,13)(9,20)(12,26,28,25)(14,24)(16,23,32,30)(19,22)(21,27,29,31)(33,34,37,42,38,43,52,60,39,44,53,61,36,41,50,58)(35,45,47,62,46,56,63,59,48,57,51,64,49,40,54,55);(1,3,6,14)(2,7,10,22)(4,12,17,28)(5,19,20,13)(8,21,15,29)(9,24,18,11)(16,31,32,27)(23,26,30,25);(1,4)(2,8)(3,12)(5,16)(6,17)(7,21)(9,23)(10,15)(11,25)(13,27)(14,28)(18,30)(19,31)(20,32)(22,29)(24,26)(33,35)(34,40)(36,49)(37,51)(38,46)(39,48)(41,57)(42,59)(43,45)(44,56)(47,53)(50,63)(52,54)(55,61)(58,62)(60,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^c=ei,f^c=fgi

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, 36893);