#435886

order	512
nilpotency class	3
isoclinism	isoclinic class of #419464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15540
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

738 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343082);