#415990

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=i,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323186);