#318760

order	512
nilpotency class	3
isoclinism	isoclinic class of #262851
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×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

546 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eghi,c^2=i,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,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, 225956);