#146791

order	512
nilpotency class	4
isoclinism	isoclinic class of #12952
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1864
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

939 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)(3,14)(4,20,23,57)(5,26)(6,31)(7,35)(8,13)(9,16)(10,22,41,21)(11,30)(12,29)(15,49,51,36)(17,48)(18,44)(19,37)(24,45,25,58)(27,39)(28,42,33,43)(32,38)(34,46)(40,54)(47,50,56,60)(52,63,55,64)(53,62,59,61);(1,3,7,17,8,19,29,18)(2,9,12,38,13,40,35,39)(4,15,24,52,23,51,25,55)(5,27,30,54,6,32,34,16)(10,36,42,63,41,49,43,64)(11,44,26,37,46,48,31,14)(20,56,58,61,57,47,45,62)(21,59,33,50,22,53,28,60);(1,4)(2,10)(3,15)(5,28)(6,33)(7,24)(8,23)(9,36)(11,45)(12,42)(13,41)(14,47)(16,53)(17,52)(18,55)(19,51)(20,26)(21,30)(22,34)(25,29)(27,60)(31,57)(32,50)(35,43)(37,56)(38,63)(39,64)(40,49)(44,62)(46,58)(48,61)(54,59)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dhi,e^b=eh,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	#72314268, #72314281
maximal normal subgroup of	#72314268, #72314281

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53987);