#457760

order	512
nilpotency class	3
isoclinism	isoclinic class of #457760
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16301
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

825 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364956);