#485830

order	512
nilpotency class	4
isoclinism	isoclinic class of #485597
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

890 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393026);