#471209

order	512
nilpotency class	3
isoclinism	isoclinic class of #462392
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2517
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16776
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

539 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378405);