#479859

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	#17284
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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=egh,c^2=f,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,g^a=gi,h^a=hi,c^b=cg,d^b=def,d^c=de,e^c=ei,g^c=gi,h^c=hi,g^d=gi,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, 387055);