#146732

order	512
nilpotency class	4
isoclinism	isoclinic class of #128583
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10788
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

935 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53928);