#482433

order	512
nilpotency class	4
isoclinism	isoclinic class of #482367
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	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1135 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240206, #72240210, #72240213, …
maximal normal subgroup of	#72240206, #72240210, #72240213, …
maximal quotient of	#72240410

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389629);