#486130

order	512
nilpotency class	4
isoclinism	isoclinic class of #486127
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×C₄⋊C₄	C₂⁴
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 871 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)(4,12)(5,7)(6,11)(8,14)(10,16)(13,15)(17,18)(19,27)(20,28)(21,24)(22,26)(23,29)(25,30)(31,32);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,12)(11,14)(18,24)(19,21)(20,22)(25,28)(26,32)(27,29);(17,19,23,21)(18,25,29,32)(20,30,22,31)(24,26,27,28);(2,7)(3,10)(8,12)(11,13)(17,20,23,22)(18,26,29,28)(19,30,21,31)(24,25,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279170
maximal normal subgroup of	#72279170

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393326);