#108447

order	512
nilpotency class	3
isoclinism	isoclinic class of #105688
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1434
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

499 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

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, 15643);