#488754

order	512
nilpotency class	4
isoclinism	isoclinic class of #488719
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

795 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 795 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395950);