#146798

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:

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

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=ehi,h^a=hi,d^b=dhi,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314266, #72314275
maximal normal subgroup of	#72314266, #72314275

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53994);