#126729

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#95
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

459 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gi,e^2=h,f^2=h,h^2=i,b^a=bd,d^a=dg,f^a=fgi,c^b=ci,e^b=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33925);