#118630

order	512
nilpotency class	3
isoclinism	isoclinic class of #10208
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

521 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=eh,f^b=fhi,d^c=di,e^c=ei,f^c=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25826);