#436353

order	512
nilpotency class	3
isoclinism	isoclinic class of #420416
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15556
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

929 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=e,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 343549);