#486147

order	512
nilpotency class	4
isoclinism	isoclinic class of #486147
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1295 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)(12,25)(13,24)(16,21)(18,28)(19,22)(20,27)(23,30)(26,32)(29,31);(1,3)(2,7)(4,16)(5,12)(6,13)(8,20)(9,21)(10,24)(11,15)(14,17)(18,29)(19,26)(22,31)(23,32)(25,28)(27,30);(2,8)(3,12)(9,22)(11,25)(13,26)(14,27)(16,29)(24,32);(2,9)(3,13)(8,22)(10,15)(12,26)(14,16)(23,28)(27,29)

Polycyclic presentation:

2^9|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	#72283632, #72284152
maximal normal subgroup of	#72283632, #72284152
maximal quotient of	#72284431

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393343);