#486925

order	512
nilpotency class	4
isoclinism	isoclinic class of #486795
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	C₂×#392	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

979 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 979 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394121);