#487204

order	512
nilpotency class	4
isoclinism	isoclinic class of #486819
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	#52548	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 911 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394400);