#105759

order	512
nilpotency class	3
isoclinism	isoclinic class of #105752
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	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

700 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:

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

Polycyclic presentation:

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

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, 12955);