#137719

order	512
nilpotency class	3
isoclinism	isoclinic class of #9239
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

536 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)(3,5,7,8)(4,6)(9,10,14,19,15,20,34,38,16,21,35,39,13,18,23,24)(11,22,26,40,25,30,37,33,27,32,17,36,31,12,28,29);(1,3,4,7)(2,5)(6,8)(9,11,23,37,13,31,35,26,16,27,34,28,15,25,14,17)(10,12,24,40,18,32,39,29,21,30,38,36,20,22,19,33);(9,12)(10,17)(11,24)(13,32)(14,33)(15,22)(16,30)(18,37)(19,25)(20,28)(21,26)(23,40)(27,38)(29,35)(31,39)(34,36)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dgi,e^b=ei,f^b=fgi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44915);