#107677

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:

(1,2,5,11,7,10,6,12)(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)(35,39,44,55)(36,46)(37,49)(38,41)(40,48)(42,52)(43,59,61,63)(45,57,50,54)(47,58)(51,56)(53,62,64,60);(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)(34,39,41,55)(36,43)(37,50)(38,44)(40,53,56,64)(42,57,49,54)(45,52)(46,63,58,59)(47,61)(48,62)(51,60);(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)(34,40)(35,43)(37,51)(38,47)(39,53)(41,56)(42,58)(44,61)(45,62)(46,49)(48,52)(50,60)(54,63)(55,64)(57,59)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,f^2=hi,g^2=hi,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, 14873);