#508055

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
prime radical	#60643	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1107 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1107 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)(11,21)(12,23)(13,18)(14,22)(15,32)(16,36)(17,26)(19,31)(20,29)(24,34)(25,28)(27,33)(30,40)(35,39)(37,38);(2,5)(6,8)(9,11)(10,12)(13,23)(14,17)(15,33)(16,37)(18,21)(19,34)(20,38)(22,27)(24,35)(25,29)(26,32)(28,36)(30,39)(31,40);(1,3)(2,6)(4,7)(5,8)(9,12,13,21)(10,17,18,33)(11,22,23,32)(14,30,15,34)(16,38,29,28)(19,37,39,25)(20,40,36,24)(26,35,27,31);(3,7)(6,8)(10,18)(12,21)(19,39)(20,36)(22,32)(26,27)(28,38)(30,34)

Polycyclic presentation:

2^9|c^2=i,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, 415251);