#120414

order	512
nilpotency class	3
isoclinism	isoclinic class of #120412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

385 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)(4,16,19,31)(5,20)(6,10)(7,11)(8,14)(9,18,26,17)(13,22,29,25)(15,23)(21,24)(27,28,30,32)(33,34,38,42,39,44,37,43)(35,40,47,55,48,56,46,60)(36,49,51,41,52,58,53,59)(45,57,61,54,62,64,50,63);(1,3,5,14,7,15,6,21)(2,8,10,12,11,24,20,23)(4,13,17,28,19,29,18,32)(9,25,16,30,26,22,31,27)(33,35)(34,40)(36,50)(37,46)(38,47)(39,48)(41,57)(42,55)(43,60)(44,56)(45,51)(49,63)(52,61)(53,62)(54,58)(59,64);(1,4,5,17,7,19,6,18)(2,9,10,16,11,26,20,31)(3,13,14,28,15,29,21,32)(8,22,12,27,24,25,23,30)(33,36)(34,41)(35,45)(37,53)(38,51)(39,52)(40,54)(42,58)(43,49)(44,59)(46,50)(47,61)(48,62)(55,64)(56,63)(57,60)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27610);