#120411

order	512
nilpotency class	3
isoclinism	isoclinic class of #120407
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	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

382 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27607);