#6342473

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	Aut(G) order 786432
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄²	C₂⁵
commutator, socle	C₂²	C₂³×C₄²
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,g^2=i,d^a=di,e^a=eh,c^b=chi,d^b=dh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209999, #72210002, #72235275, …
maximal normal subgroup of	#72209999, #72210002, #72235275, …
maximal quotient of	C₂×#6342473, #72239964, #72240058

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249669);