#487557

order	512
nilpotency class	4
isoclinism	isoclinic class of #486827
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 ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

867 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=g,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=dgi,e^c=eh,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284971, #72285126, #72285376
maximal normal subgroup of	#72284971, #72285126, #72285376

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394753);