#323632

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

656 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230828);