#320867

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, 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

638 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)(11,17)(12,15)(13,16)(14,19)(18,22)(20,24)(21,23)(25,26,31,27,29,33,30,34)(28,35,38,32,36,40,37,39);(2,5)(6,7)(9,11,13,20,12,19,21,18)(10,14,16,22,15,17,23,24)(25,27,31,33,29,34,30,26)(28,32,38,40,36,39,37,35);(10,15)(11,18)(13,21)(14,22)(17,24)(19,20)(25,28,29,36)(26,32,33,39)(27,35,34,40)(30,38,31,37);(1,3,4,8)(2,6,5,7)(10,15)(14,17)(16,23)(22,24)(26,33)(27,34)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,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, 228063);