#147254

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1879
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2,4,3)(5,6)(7,20)(8,26)(9,17)(10,16)(11,18)(12,34)(13,19)(14,22)(15,28)(21,51)(23,40)(24,38)(25,50)(27,44)(29,42)(30,62)(31,39)(32,47)(33,48)(35,46)(36,61)(37,54)(41,58)(43,63)(45,64)(49,53)(52,60)(55,67)(56,66)(57,65)(59,68);(5,7,11,24,13,25,12,35)(6,14,18,39,19,41,34,40)(8,21,29,54,30,56,36,55)(9,31,32,58,33,23,10,22)(15,37,43,66,45,67,44,51)(16,46,17,20,47,38,48,50)(26,60,42,49,62,68,61,65)(27,57,28,52,63,53,64,59);(1,3,4,2)(5,8)(6,15)(7,21)(9,28)(10,27)(11,29)(12,36)(13,30)(14,37)(16,26)(17,42)(18,43)(19,45)(20,49)(22,57)(23,59)(24,54)(25,56)(31,52)(32,63)(33,64)(34,44)(35,55)(38,68)(39,66)(40,51)(41,67)(46,60)(47,62)(48,61)(50,65)(53,58)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54450);