#107675

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	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

671 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)(4,17,19,24,18,26,21,9)(14,28,30,22,29,32,31,27)(33,34,37,42,38,41,51,56)(35,43,46,59,45,39,52,54)(36,47,48,57,49,40,50,55)(44,53,61,63,60,58,62,64);(1,3,6,15,7,16,5,20)(2,8,12,23,10,13,11,25)(4,14,19,31,18,29,21,30)(9,22,17,28,24,27,26,32)(33,35)(34,39,41,43)(36,44)(37,46)(38,45)(40,53,47,58)(42,54,56,59)(48,61)(49,60)(50,62)(51,52)(55,63,57,64);(1,4)(2,9)(3,14)(5,21)(6,19)(7,18)(8,22)(10,24)(11,26)(12,17)(13,27)(15,31)(16,29)(20,30)(23,28)(25,32)(33,36,38,49)(34,40,41,47)(35,44,45,60)(37,50,51,48)(39,53,43,58)(42,57,56,55)(46,62,52,61)(54,64,59,63)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,e^2=h,f^2=hi,g^2=h,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, 14871);