#490546

order	512
nilpotency class	4
isoclinism	isoclinic class of #490474
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

448 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eg,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^c=di,f^c=fh

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, 397742);