#473746

order	512
nilpotency class	3
isoclinism	isoclinic class of #457096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 380942);