#486356

order	512
nilpotency class	4
isoclinism	isoclinic class of #486147
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

855 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 855 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,c^b=ci,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282351, #72282469, #72282764
maximal normal subgroup of	#72282351, #72282469, #72282764

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393552);