#474190

order	512
nilpotency class	3
isoclinism	isoclinic class of #469927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2605
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16905
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

565 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)(3,13,17,8)(4,18)(5,10)(6,12)(7,11)(9,20)(14,30,31,26)(15,24,16,25)(19,29,32,23)(21,27)(22,28);(1,3)(2,8)(4,19,22,32)(5,16)(6,15)(7,17)(9,26,27,30)(10,24)(11,13)(12,25)(14,21,31,20)(18,23,28,29);(1,4,5,20,7,22,6,21)(2,9,12,18,11,27,10,28)(3,14,15,19,17,31,16,32)(8,23,24,26,13,29,25,30);(2,10,11,12)(8,24,13,25)(9,18,27,28)(23,30,29,26)

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=f,d^2=e,f^2=i,g^2=i,h^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,f^b=fi,d^c=de,h^c=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, 381386);