#108970

order	512
nilpotency class	3
isoclinism	isoclinic class of #8626
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

826 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)(3,7,8,5)(4,6)(9,10)(11,19)(12,23)(13,26)(14,18)(15,21)(16,24)(17,27)(20,33)(22,29)(25,38)(28,40)(30,39)(31,35)(32,36)(34,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15,18,19)(12,20)(13,22)(14,21)(16,28,30,31)(17,29,26,32)(23,37,38,33)(24,35)(25,34)(27,36)(39,40);(9,12,13,24,14,25,27,39)(10,16,17,23,18,30,26,38)(11,20,22,35,21,34,36,40)(15,28,29,37,19,31,32,33)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=di,f^b=fh

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, 16166);