#505729

order	512
nilpotency class	4
isoclinism	isoclinic class of #500887
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

557 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 557 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=ei,d^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,f^c=fhi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412925);