#459921

order	512
nilpotency class	3
isoclinism	isoclinic class of #459921
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16352
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

941 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:

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

Polycyclic presentation:

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

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, 367117);