#276950

order	512
nilpotency class	3
isoclinism	isoclinic class of #268891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

502 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)(9,10,13,17,12,16,21,15)(11,18,20,23,19,14,22,24)(27,34)(31,39)(35,40)(36,37);(2,5)(6,7)(9,11)(10,14)(12,19)(13,20)(15,23)(16,18)(17,24)(21,22)(25,26,29,33)(27,35,34,40)(28,32,30,38)(31,36,39,37);(10,15)(13,21)(14,23)(16,17)(18,24)(20,22)(25,27,29,34)(26,31,33,39)(28,37,30,36)(32,40,38,35);(1,3,4,8)(2,6,5,7)(10,16)(11,19)(15,17)(20,22)(25,28,29,30)(26,32,33,38)(27,36,34,37)(31,35,39,40)

Polycyclic presentation:

2^9|a^2=efhi,b^2=i,c^2=i,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,g^c=gi

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, 184146);