#486300

order	512
nilpotency class	4
isoclinism	isoclinic class of #486299
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#62244	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

912 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=g,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393496);