#119719

order	512
nilpotency class	3
isoclinism	isoclinic class of #119716
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

692 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,e^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ehi,g^b=gh,e^c=ei,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26915);