#149297

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1954
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56493);