#484626

order	512
nilpotency class	4
isoclinism	isoclinic class of #484199
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	#33517	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

959 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 959 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391822);