#482748

order	512
nilpotency class	4
isoclinism	isoclinic class of #482722
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#2540	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#556 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

850 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)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,21,26,22,27,19,24)(20,25,28,32,29,23,30,31);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(18,23)(20,29)(24,31)(25,27)(26,32)(28,30);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,19,22,21)(18,24,27,26)(20,28,29,30)(23,31,25,32);(3,10)(7,14)(11,16)(13,15)(17,19,22,21)(18,24,27,26)(20,28,29,30)(23,31,25,32)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=dhi,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389944);