#107364

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#120
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

701 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)(3,11,14,7)(4,15,16,26)(5,10)(6,9)(8,17,23,19)(12,28)(13,25,18,22)(20,24)(21,30)(27,31)(29,32)(33,34,37,42,39,44,38,43)(35,45,47,54,48,58,52,40)(36,41,50,56,51,57,49,55)(46,60,63,53,62,59,61,64);(1,3)(2,7)(4,12)(5,18)(6,14)(8,21)(9,11)(10,25)(13,20)(15,28)(16,29)(17,31)(19,30)(22,24)(23,27)(26,32)(33,35,38,47,39,48,37,52)(34,40,43,45,44,54,42,58)(36,46,49,63,51,62,50,61)(41,53,55,59,57,64,56,60);(1,4)(2,8)(3,12)(5,19)(6,16)(7,21)(9,23)(10,26)(11,27)(13,31)(14,29)(15,24)(17,20)(18,30)(22,28)(25,32)(33,36)(34,41)(35,46)(37,50)(38,49)(39,51)(40,53)(42,56)(43,55)(44,57)(45,59)(47,63)(48,62)(52,61)(54,64)(58,60)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14560);