#470891

order	512
nilpotency class	3
isoclinism	isoclinic class of #457608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2735
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16708
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

717 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:

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

Polycyclic presentation:

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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378087);