#321908

order	512
nilpotency class	3
isoclinism	isoclinic class of #321901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

720 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 186 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^c=fi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229104);