#452814

order	512
nilpotency class	3
isoclinism	isoclinic class of #427311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16065
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

585 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)(3,14,17,9)(4,18,21,28)(5,22,16,25)(6,12)(7,13)(8,11)(10,20,27,23)(15,30)(19,24)(26,32)(29,31);(1,3,7,16,8,17,6,5)(2,9,13,22,11,14,12,25)(4,19,20,32,21,31,23,15)(10,26,18,29,27,30,28,24);(1,4)(2,10,11,27)(3,15)(5,19)(6,20)(7,23)(8,21)(9,24,14,29)(12,18,13,28)(16,31)(17,32)(22,30,25,26);(1,5,6,17,8,16,7,3)(2,9,12,25,11,14,13,22)(4,19,20,32,21,31,23,15)(10,24,18,26,27,29,28,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=e,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^b=hi,g^c=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, 360010);