#488753

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

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

normal subgroups only

This group has 790 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)(9,10)(11,23)(12,16)(13,21)(14,20)(15,22)(17,25)(18,29)(19,27)(24,34)(26,37)(28,35)(30,33)(31,36)(32,40)(38,39);(1,2)(3,8)(4,6)(5,7)(10,16)(12,27)(13,14)(17,32)(18,20)(19,21)(22,33)(23,34)(24,39)(25,31)(28,40)(29,30)(35,37)(36,38);(3,7)(5,8)(9,11,15,26)(10,17,22,36)(12,28,30,24)(13,31,14,25)(16,32,33,38)(18,35,19,34)(20,37,21,23)(27,40,29,39);(2,6)(5,8)(9,12,13,29,15,30,14,27)(10,18,20,16,22,19,21,33)(11,24,31,40,26,28,25,39)(17,34,37,38,36,35,23,32)

Polycyclic presentation:

2^9|c^2=i,d^2=gh,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, 395949);