#146771

order	512
nilpotency class	4
isoclinism	isoclinic class of #12750
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1786
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 ◁ C₂³×C₄ ◁ 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:

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

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,h^a=hi,d^b=dhi,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,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, 53967);