#286382

order	512
nilpotency class	3
isoclinism	isoclinic class of #282171
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=e,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 193578);