#489186

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
prime radical	#33717	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 793 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:

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

Polycyclic presentation:

2^9|b^2=i,c^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, 396382);