#453878

order	512
nilpotency class	3
isoclinism	isoclinic class of #427666
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2527
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16190
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

546 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)(3,7,13,22)(4,9,15,26)(5,8)(6,10)(11,18)(12,20)(14,19,17,21)(16,23,29,25)(24,28)(27,31)(30,32);(1,3)(2,7,10,22)(4,12,15,28)(5,17)(6,13)(8,21,11,19)(9,24)(14,18)(16,30,29,27)(20,26)(23,31)(25,32);(2,8,10,11)(7,19,22,21)(9,23,26,25)(20,31,24,32);(1,4,5,16,6,15,18,29)(2,9,11,25,10,26,8,23)(3,12,14,27,13,28,17,30)(7,20,21,32,22,24,19,31)

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=g,d^2=f,f^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=de,e^b=ei,f^b=fi,e^d=ei,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, 361074);