#180212

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
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

677 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87408);