#486151

order	512
nilpotency class	4
isoclinism	isoclinic class of #486147
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
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1263 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241080
maximal normal subgroup of	#72241080

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393347);