#286610

order	512
nilpotency class	3
isoclinism	isoclinic class of #281915
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

553 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)(9,10,14,19,13,18,12,17)(11,20,21,24,22,15,23,16)(25,26,31,37,30,28,29,33)(27,32,38,40,34,36,35,39);(2,5)(6,8)(9,11,14,21,13,22,12,23)(10,15,19,16,18,20,17,24)(25,27,31,38,30,34,29,35)(26,32,37,40,28,36,33,39);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,19)(17,20)(18,24)(25,28,30,26)(27,32,34,36)(29,33,31,37)(35,40,38,39);(2,5)(6,8)(9,12,13,14)(10,17,18,19)(11,21,22,23)(15,16,20,24)(26,28)(27,34)(33,37)(35,38)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=eh,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=dh,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, 193806);