#479794

order	512
nilpotency class	3
isoclinism	isoclinic class of #477835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2793
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17253
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

522 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eh,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,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, 386990);