#508057

order	512
nilpotency class	4
isoclinism	isoclinic class of #508016
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
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

1097 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1097 conjugacy classes of subgroups — too many to draw. Only its 152 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)(9,10)(11,16)(12,17)(13,30)(14,31)(15,20)(18,27)(19,28)(21,39)(22,35)(23,40)(24,33)(25,37)(26,34)(29,38)(32,36);(2,5)(6,8)(9,11,15,25)(10,16,20,37)(12,26,29,23)(13,21,27,22)(14,32,28,24)(17,34,38,40)(18,35,30,39)(19,33,31,36);(1,3)(2,6)(4,7)(5,8)(10,17)(11,21)(12,27)(13,29)(14,28)(16,33)(18,19)(20,38)(22,25)(23,32)(24,26)(30,31)(35,39)(36,37);(3,7)(6,8)

Polycyclic presentation:

2^9|b^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,d^c=de,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415253);