#149325

order	512
nilpotency class	4
isoclinism	isoclinic class of #13331
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#459
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,25)(16,22)(17,21)(18,33)(19,23)(20,26)(24,30)(27,41)(28,35)(29,42)(31,40)(32,39)(34,36)(37,43)(38,44);(5,7,9,8)(6,10,11,12)(13,15,19,29)(14,20,23,37)(16,27,31,44)(17,26,32,43)(18,34,30,28)(21,36,39,35)(22,25,40,42)(24,41,33,38);(5,8,9,7)(6,10,11,12)(14,21)(16,17)(18,30)(20,35)(22,33)(23,39)(24,40)(25,41)(26,27)(28,34)(31,32)(36,37)(38,42)(43,44)

Polycyclic presentation:

2^9|a^2=g,b^2=dei,c^2=de,d^2=hi,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,h^b=hi,d^c=dh,e^c=ehi,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, 56521);