#426746

order	512
nilpotency class	3
isoclinism	isoclinic class of #417686
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15358
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1013 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)(3,11,14,26)(4,8)(5,9)(6,10)(7,13,21,18)(12,29,31,28)(15,23)(16,25)(17,24)(19,27)(20,32,22,30);(1,3,5,18,6,14,19,13)(2,7,9,26,10,21,27,11)(4,12,15,32,17,31,16,30)(8,22,23,28,24,20,25,29);(1,4,5,15,6,17,19,16)(2,8,9,23,10,24,27,25)(3,12,18,32,14,31,13,30)(7,20,26,29,21,22,11,28);(1,3)(2,7)(4,12)(5,18)(6,14)(8,20)(9,26)(10,21)(11,27)(13,19)(15,32)(16,30)(17,31)(22,24)(23,29)(25,28)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,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, 333942);