#478638

order	512
nilpotency class	3
isoclinism	isoclinic class of #476143
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2819
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17158
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

657 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385834);