#391966

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2826
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14666
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

868 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

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, 299162);