#172874

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

807 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=h,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, 80070);