#508054

order	512
nilpotency class	4
isoclinism	isoclinic class of #508024
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 ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1128 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 150 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)(13,28)(14,30)(15,18)(19,26)(20,27)(21,33)(22,40)(23,38)(24,32)(25,34)(29,36)(31,35)(37,39);(2,5)(6,8)(9,11)(10,16)(12,25)(13,21)(14,31)(15,24)(17,34)(18,32)(19,37)(20,38)(22,29)(23,27)(26,39)(28,33)(30,35)(36,40);(1,3)(2,6)(4,7)(5,8)(10,17)(11,21)(12,26)(13,29)(14,27)(18,28)(19,20)(22,23)(24,25)(30,36)(31,39)(33,40)(34,37)(35,38);(3,7)(6,8)(10,18)(16,32)(17,28)(19,36)(20,30)(33,34)(35,38)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415250);