#129536

order	512
nilpotency class	3
isoclinism	isoclinic class of #8412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1779
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

594 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dgi,f^b=fi,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, 36732);