#473267

order	512
nilpotency class	3
isoclinism	isoclinic class of #473267
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16857
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

692 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)(3,12,13,7)(4,16)(6,8,19,11)(9,18)(14,25)(15,20,27,22)(17,24)(21,26)(23,29)(28,30)(31,32);(1,3,6,15,5,13,19,27)(2,7,8,22,10,12,11,20)(4,14,17,28,18,26,29,32)(9,21,23,31,16,25,24,30);(2,8)(3,13)(6,19)(7,20)(9,23)(10,11)(12,22)(14,26)(16,24)(17,29)(21,30)(25,31);(1,4,6,17,5,18,19,29)(2,9,11,24,10,16,8,23)(3,14,15,28,13,26,27,32)(7,21,20,30,12,25,22,31)

Polycyclic presentation:

2^9|a^2=e,b^2=efgh,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,e^d=ei,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, 380463);