#180408

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

650 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)(11,15)(12,17)(13,22)(14,18)(16,19)(20,24)(21,23)(25,26,28,31)(27,33,35,40)(29,37,36,32)(30,38,39,34);(1,2,5,8)(3,6,4,7)(9,11)(10,15)(12,21)(13,19)(14,20)(16,22)(17,23)(18,24)(25,26)(27,30)(28,31)(29,37)(32,36)(33,38)(34,40)(35,39);(3,4)(6,7)(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,24)(18,23)(25,27)(26,30)(28,35)(29,38)(31,39)(32,40)(33,37)(34,36);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,17,18,22)(11,19,20,21)(15,16,24,23)(26,31)(30,39)(32,37)(33,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=hi,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, 87604);