#107814

order	512
nilpotency class	3
isoclinism	isoclinic class of #107810
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	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

678 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)(4,8)(5,17)(6,10)(7,13)(9,19)(12,27)(14,26)(15,24)(16,23)(18,22)(20,28)(21,31)(25,29)(30,32)(33,34)(35,44)(36,49,51,57)(37,42)(38,53)(39,43)(40,46)(41,50,56,52)(45,60,61,64)(47,59)(48,55)(54,63,58,62);(1,3)(2,7,10,21)(4,12)(5,13)(6,14)(8,20,23,32)(9,11,17,26)(15,28)(16,29)(18,30)(19,31)(22,27,24,25)(33,35,38,48,39,46,37,47)(34,40,42,55,43,44,53,59)(36,45,50,62,51,61,52,63)(41,54,57,60,56,58,49,64);(1,4,6,16)(2,8,10,23)(3,12,14,29)(5,18,19,15)(7,20,21,32)(9,24,17,22)(11,25,26,27)(13,30,31,28)(33,36)(34,41)(35,45)(37,52)(38,50)(39,51)(40,54)(42,57)(43,56)(44,58)(46,61)(47,63)(48,62)(49,53)(55,60)(59,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di,e^b=eh,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15010);