#482848

order	512
nilpotency class	4
isoclinism	isoclinic class of #482734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×#546 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

974 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,b^a=be,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fghi,d^c=dh,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390044);