#287390

order	512
nilpotency class	3
isoclinism	isoclinic class of #282043
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

568 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=f,c^2=h,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,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, 194586);