#170119

order	512
nilpotency class	3
isoclinism	isoclinic class of #169613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

677 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,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=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 77315);