#320993

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
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×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

632 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,23)(19,22)(20,24)(25,26,29,34,31,33,30,27)(28,35,36,39,38,40,37,32);(2,5)(6,7)(9,11,12,18,13,19,20,17)(10,14,16,23,15,22,24,21)(25,27,30,33,31,34,29,26)(28,32,37,40,38,39,36,35);(11,17)(12,20)(14,21)(16,24)(18,19)(22,23)(25,28,31,38)(26,32,33,39)(27,35,34,40)(29,37,30,36);(1,3,4,8)(2,6,5,7)(10,15)(14,22)(16,24)(21,23)(26,33)(27,34)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228189);