#331653

order	512
nilpotency class	3
isoclinism	isoclinic class of #331304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

612 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238849);