#479786

order	512
nilpotency class	3
isoclinism	isoclinic class of #476443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2829
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17246
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

501 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)(3,12)(4,17)(5,19,21,9)(6,22)(7,11)(8,15)(10,13)(14,25,30,24)(16,27)(18,31,29,23)(20,26,28,32);(1,3,13,15,7,16,6,4)(2,8,10,27,11,17,22,12)(5,18,20,30,21,29,28,14)(9,24,32,23,19,25,26,31);(1,4,6,16,7,15,13,3)(2,8,10,27,11,17,22,12)(5,14,20,18,21,30,28,29)(9,25,26,23,19,24,32,31);(1,5,6,20,7,21,13,28)(2,9,10,26,11,19,22,32)(3,14,15,29,16,30,4,18)(8,23,12,25,17,31,27,24)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386982);