#512102

order	512
nilpotency class	4
isoclinism	isoclinic class of #511753
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	C₂²×C₈	C₂⁴
prime radical	#33520	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

990 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

2^9|c^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,h^a=hi,d^b=di,f^c=fh,g^c=ghi,h^c=hi,f^d=fhi,g^d=gh,h^d=hi

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, 419298);