#169018

order	512
nilpotency class	3
isoclinism	isoclinic class of #168909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

(1,2,4,7)(3,5,8,6)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24)(27,34)(28,29)(31,38)(32,36)(35,37)(39,40);(9,11,14,21)(10,15,17,23)(12,20,13,19)(16,24,22,18)(25,26)(27,31)(28,32)(29,36)(30,33)(34,39)(35,40)(37,38);(2,5)(3,8)(6,7)(25,27,30,35)(26,31,33,40)(28,34,29,37)(32,39,36,38);(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)(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=g,c^2=h,d^2=gh,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, 76214);