#467399

order	512
nilpotency class	3
isoclinism	isoclinic class of #455576
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16568
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

771 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:

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

Polycyclic presentation:

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

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, 374595);