#107519

order	512
nilpotency class	3
isoclinism	isoclinic class of #107494
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

688 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:

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

Polycyclic presentation:

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

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, 14715);