#489004

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

767 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 767 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396200);