#477838

order	512
nilpotency class	3
isoclinism	isoclinic class of #477838
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

703 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385034);