#471304

order	512
nilpotency class	3
isoclinism	isoclinic class of #457096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

816 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)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(18,23)(19,27)(20,21)(24,31)(25,26)(28,32);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,18,20,23,22,25,21,26)(19,28,30,24,27,31,29,32);(2,5)(3,7)(9,11,13,12)(10,15,16,14)(17,19,20,29,22,27,21,30)(18,24,23,28,25,32,26,31);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(18,25)(19,29,27,30)(23,26)(24,31,32,28)

Polycyclic presentation:

2^9|a^2=eg,b^2=efh,c^2=ef,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=di,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=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, 378500);