#6432475

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2076 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,h^b=hi,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239646, #72313961, #72314032
maximal normal subgroup of	#72239646, #72313961, #72314032
maximal quotient of	#72239782

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339671);