#490577

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

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

464 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,5)(4,6)(7,8)(9,10,15,19)(11,20,17,21)(12,24,25,35)(13,29,31,23)(14,32,30,18)(16,28,33,27)(22,26,34,40)(36,39,37,38);(2,5)(6,7)(9,11,21,10,14,23,13,18,15,17,20,19,30,29,31,32)(12,22,37,16,28,39,26,35,25,34,36,33,27,38,40,24);(2,6)(4,8)(5,7)(9,12)(10,16)(11,22)(13,26)(14,28)(15,25)(17,34)(18,35)(19,33)(20,36)(21,37)(23,39)(24,32)(27,30)(29,38)(31,40);(12,25)(16,33)(22,34)(24,35)(26,40)(27,28)(36,37)(38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=egh,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, 397773);