#345137

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
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×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,8)(3,4,7,6)(9,10,12,16,13,17,20,15)(11,14,18,22,19,23,24,21)(25,26,29,31)(27,32,35,38)(28,36,30,33)(34,40,37,39);(1,3,5,7)(2,6,8,4)(9,11,13,19)(10,14,17,23)(12,18,20,24)(15,21,16,22)(26,31)(27,34)(28,30)(32,39)(35,37)(38,40);(1,4,5,6)(2,7,8,3)(10,15)(12,20)(14,21)(16,17)(18,24)(22,23)(25,27,29,35)(26,32,31,38)(28,34,30,37)(33,39,36,40);(2,8)(3,7)(10,16)(12,20)(14,22)(15,17)(18,24)(21,23)(25,28,29,30)(26,33,31,36)(27,34,35,37)(32,39,38,40)

Polycyclic presentation:

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

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, 252333);