#481857

order	512
nilpotency class	4
isoclinism	isoclinic class of #481819
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	C₂⁶	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1420 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)(3,11,14,29)(4,16,19,9)(5,8)(6,10)(7,13,21,17)(12,27,15,25)(18,24)(20,26,32,23)(22,30,28,31);(1,3)(2,7,10,22)(4,17)(5,15)(6,12)(8,21,24,28)(9,25,16,29)(11,26,27,23)(13,20)(14,18)(19,31)(30,32);(2,8)(7,21)(9,23)(10,24)(11,25)(16,26)(22,28)(27,29);(3,12)(7,22)(11,27)(13,30)(14,15)(17,31)(21,28)(25,29)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284343, #72284432, #72284476, …
maximal normal subgroup of	#72284343, #72284432, #72284476, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389053);