#107670

order	512
nilpotency class	3
isoclinism	isoclinic class of #107658
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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

695 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)(33,34,37,43,39,42,38,53)(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)(2,7)(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)(33,35,38,48,39,46,37,47)(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)(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)(35,45)(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,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fhi

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, 14866);