#323619

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

Polycyclic presentation:

2^9|a^2=ef,b^2=egi,c^2=h,d^2=ei,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, 230815);