#483278

order	512
nilpotency class	4
isoclinism	isoclinic class of #482975
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#62476	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

683 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 683 conjugacy classes of subgroups — too many to draw. Only its 238 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=hi,f^2=gh,h^2=i,b^a=be,c^a=ci,d^a=di,e^a=eg,e^b=eh,f^b=fgh,h^b=hi,d^c=dgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390474);