#172862

order	512
nilpotency class	3
isoclinism	isoclinic class of #172777
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#39444	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,d^b=dh,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, 80058);