#505674

order	512
nilpotency class	4
isoclinism	isoclinic class of #500676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
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

591 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 591 conjugacy classes of subgroups — too many to draw. Only its 158 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)(11,19)(12,24)(13,28)(14,30)(15,20)(16,22)(17,21)(18,29)(23,37)(25,36)(26,40)(27,32)(31,33)(34,38)(35,39);(1,3)(2,5)(4,7)(6,8)(9,11,13,22,15,23,29,14)(10,16,18,19,20,30,28,37)(12,21,25,38,27,39,26,31)(17,32,33,36,35,24,34,40);(3,7)(5,8)(9,12)(10,17)(11,21)(13,25)(14,31)(15,27)(16,32)(18,33)(19,36)(20,35)(22,38)(23,39)(24,30)(26,29)(28,34)(37,40);(9,11,13,22,15,23,29,14)(10,16,18,19,20,30,28,37)(12,21,25,38,27,39,26,31)(17,32,33,36,35,24,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=e,e^2=i,f^2=hi,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=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238786, #72239811, #72239840
maximal normal subgroup of	#72238786, #72239811, #72239840

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412870);