#508229

order	512
nilpotency class	4
isoclinism	isoclinic class of #508022
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	#60676	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 769 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:

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

Polycyclic presentation:

2^9|c^2=e,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,f^c=fh,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, 415425);