#106303

order	512
nilpotency class	3
isoclinism	isoclinic class of #1909
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1155 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,e^b=eh,d^c=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13499);