#331536

order	512
nilpotency class	3
isoclinism	isoclinic class of #331114
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×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

650 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238732);