#462043

order	512
nilpotency class	3
isoclinism	isoclinic class of #455563
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	#16414
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

721 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=f,e^2=i,f^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,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,h^d=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, 369239);