#453854

order	512
nilpotency class	3
isoclinism	isoclinic class of #425083
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2576
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16158
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

701 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361050);