#313589

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
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	C₂×C₄²	C₂⁴
prime radical	C₂×D₄²	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

864 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)(3,6)(4,5)(7,8)(17,18)(19,29)(20,24)(21,25)(22,32)(23,28)(26,31)(27,30);(2,5)(6,8)(10,13)(11,15)(14,16)(17,19,22,20,23,21,30,31)(18,24,27,29,28,26,32,25);(1,3,4,7)(2,6,5,8)(9,10,12,14)(11,16,15,13)(17,20,30,19,23,31,22,21)(18,25,32,26,28,29,27,24);(2,5)(6,8)(17,21,22,31,23,19,30,20)(18,26,27,25,28,24,32,29)

Polycyclic presentation:

2^9|b^2=f,c^2=ehi,d^2=f,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 220785);