#487628

order	512
nilpotency class	4
isoclinism	isoclinic class of #487627
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
prime radical	#12090	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

983 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)(3,12)(4,7)(5,10)(6,11)(8,15)(9,17)(13,24)(14,23)(16,22)(18,21)(19,26)(20,28)(25,30)(27,32)(29,31);(2,7)(4,17)(8,20)(9,10)(11,21)(12,23)(14,28)(18,30)(22,31)(24,32)(25,26)(27,29);(3,13)(4,18)(8,22)(12,24)(14,28)(15,16)(17,30)(20,31)(23,32)(27,29);(1,3,6,16)(2,8,11,24)(4,14,18,29)(5,15,19,13)(7,20,21,32)(9,23,25,31)(10,12,26,22)(17,28,30,27)

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,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278134
maximal normal subgroup of	#72278134

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394824);