#287507

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
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

580 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 194703);