#107734

order	512
nilpotency class	3
isoclinism	isoclinic class of #107718
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

687 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,d^b=di,e^b=ehi,d^c=dhi,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, 14930);