#486104

order	512
nilpotency class	4
isoclinism	isoclinic class of #486091
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×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

925 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)(3,9)(4,12)(5,7)(6,11)(8,14)(10,16)(13,15)(17,18)(19,24)(20,28)(21,23)(22,26)(25,30)(27,31)(29,32);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,12)(11,14)(18,23)(19,20)(21,30)(24,32)(25,26)(28,31);(17,19)(18,24)(20,22)(21,29)(23,31)(25,32)(26,28)(27,30);(2,7)(3,10)(8,12)(11,13)(17,20,22,19)(18,24,26,28)(21,29,30,27)(23,31,25,32)

Polycyclic presentation:

2^9|d^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,d^c=di,e^c=eh,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278658
maximal normal subgroup of	#72278658

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393300);