#486316

order	512
nilpotency class	4
isoclinism	isoclinic class of #486312
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	#24281	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 930 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393512);