#147293

order	512
nilpotency class	4
isoclinism	isoclinic class of #12933
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1853
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10908
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

513 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)(3,15,19,46)(4,21,24,45)(5,26)(6,31)(7,34)(8,13)(9,17,39,32)(10,23,42,28)(11,29)(12,30)(14,33)(16,50)(18,47,27,40)(20,38,35,44)(22,48,59,43)(25,41,36,58)(37,53)(49,55)(51,64)(52,61)(54,63)(56,62)(57,60);(1,3)(2,9,13,39)(4,16)(5,27)(6,32)(7,35)(8,19)(10,37,42,62)(11,44,31,38)(12,46,26,15)(14,47,34,40)(17,29)(18,30)(20,33)(21,57,45,64)(22,52)(23,51)(24,54)(25,56)(28,60)(36,53)(41,61,58,49)(43,63,48,50)(55,59);(1,4,7,36,8,24,33,25)(2,10,14,48,13,42,34,43)(3,16,35,53,19,54,20,56)(5,28,29,59,30,23,6,22)(9,37,47,50,39,62,40,63)(11,45,26,58,31,21,12,41)(15,49,38,57,46,61,44,64)(17,55,18,51,32,52,27,60)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314209
maximal normal subgroup of	#72314209

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54489);