#135838

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1850
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dg,e^b=ei,d^c=di,e^c=ei

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, 43034);