#479783

order	512
nilpotency class	3
isoclinism	isoclinic class of #478348
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2754
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17244
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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,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, 386979);