#169125

order	512
nilpotency class	3
isoclinism	isoclinic class of #168909
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

657 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,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=dg,e^b=eg,f^b=fh,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, 76321);