#166256

order	512
nilpotency class	3
isoclinism	isoclinic class of #165873
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

820 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)(3,6)(4,7)(5,8)(9,10,13,15)(11,17,19,24)(12,20,22,16)(14,21,23,18)(25,26,30,29)(27,31,35,34)(28,32,37,36)(33,39,40,38);(2,6)(3,4)(7,8)(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17)(25,27,30,35)(26,31,29,34)(28,33,37,40)(32,38,36,39);(2,7)(3,4)(6,8)(25,28)(26,32)(27,33)(29,36)(30,37)(31,38)(34,39)(35,40);(2,8)(6,7)(10,15)(14,23)(16,20)(17,24)(25,29,30,26)(27,34,35,31)(28,36,37,32)(33,39,40,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,e^b=eg,f^b=fh,e^c=eh,f^c=fi

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, 73452);