#107587

order	512
nilpotency class	3
isoclinism	isoclinic class of #107562
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

673 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=di,f^b=fh,e^c=eh,f^c=fhi

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, 14783);