#107326

order	512
nilpotency class	3
isoclinism	isoclinic class of #107314
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

697 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)(3,12)(4,16,19,26)(5,20)(6,10)(7,11)(8,14)(9,18,25,17)(13,28,30,22)(15,23)(21,24)(27,29,32,31)(33,34,37,42,39,43,38,52)(35,44,47,58,46,40,48,54)(36,41,50,56,49,55,51,62)(45,53,60,63,59,57,61,64);(1,3)(2,8,11,24)(4,13)(5,14)(6,21)(7,15)(9,22,25,28)(10,12,20,23)(16,32,26,27)(17,29)(18,31)(19,30)(33,35,38,48,39,46,37,47)(34,40,42,54,43,44,52,58)(36,45,51,61,49,59,50,60)(41,53,56,63,55,57,62,64);(1,4)(2,9)(3,13)(5,18)(6,17)(7,19)(8,22)(10,26)(11,25)(12,27)(14,31)(15,30)(16,20)(21,29)(23,32)(24,28)(33,36)(34,41)(35,45)(37,50)(38,51)(39,49)(40,53)(42,56)(43,55)(44,57)(46,59)(47,60)(48,61)(52,62)(54,63)(58,64)

Polycyclic presentation:

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

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, 14522);