#493240

order	512
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

816 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 338 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=fi,e^2=hi,h^2=i,b^a=be,c^a=ci,d^a=di,e^a=eh,h^a=hi,e^b=eh,h^b=hi,d^c=di

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, 400436);