#487987

order	512
nilpotency class	4
isoclinism	isoclinic class of #487643
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	#57213	C₂
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

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

normal subgroups only

This group has 992 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)(3,12,16,8)(4,7,17,9)(6,11,19,27)(13,23,14,25)(15,24,31,20)(18,21,32,26)(22,29,28,30);(2,7)(3,13)(4,17)(8,22)(9,10)(11,21)(12,28)(14,16)(15,30)(18,32)(20,25)(23,24)(26,27)(29,31);(3,14)(4,18)(8,23)(12,25)(13,16)(15,31)(17,32)(20,28)(22,24)(29,30);(1,3,5,16)(2,8,10,12)(4,15,17,31)(6,13,19,14)(7,20,9,24)(11,25,27,23)(18,29,32,30)(21,22,26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395183);