#120444

order	512
nilpotency class	3
isoclinism	isoclinic class of #10508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1441
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

400 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=i,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,f^b=fhi,e^c=ehi,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, 27640);