#296382

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

819 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 203578);