#387223

order	512
nilpotency class	3
isoclinism	isoclinic class of #32696
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2749
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14456
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

784 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294419);