#478446

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	#17148
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

790 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,f^c=fi,h^c=hi,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, 385642);