#457984

order	512
nilpotency class	3
isoclinism	isoclinic class of #454109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16307
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

768 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,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, 365180);