#107313

order	512
nilpotency class	3
isoclinism	isoclinic class of #8491
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

673 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)(3,13,16,8)(4,18,20,26)(5,11)(6,12)(7,10)(9,19,25,21)(14,28)(15,24,17,23)(22,29)(27,31)(30,32)(33,34,38,42,39,44,37,43)(35,45,47,40,48,55,53,56)(36,49,51,41,52,57,50,58)(46,60,61,64,62,54,63,59);(1,3,6,17,7,16,5,15)(2,8,12,24,10,13,11,23)(4,14,19,29,20,30,21,31)(9,22,18,32,25,27,26,28)(33,35)(34,40)(36,46)(37,53)(38,47)(39,48)(41,54)(42,55)(43,45)(44,56)(49,64)(50,63)(51,61)(52,62)(57,59)(58,60);(1,4,7,20)(2,9,10,25)(3,14,16,30)(5,21,6,19)(8,22,13,27)(11,26,12,18)(15,31,17,29)(23,28,24,32)(33,36,39,52)(34,41,44,58)(35,46,48,62)(37,51,38,50)(40,54,56,60)(42,49,43,57)(45,59,55,64)(47,63,53,61)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=ei,f^b=fhi,d^c=di,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, 14509);