#486315

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	#62270	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) = 128

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

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

Polycyclic presentation:

2^9|a^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, 393511);