#297008

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
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	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

713 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:

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

Polycyclic presentation:

2^9|a^2=ehi,c^2=fh,d^2=ehi,f^2=h,g^2=i,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^b=gi,d^c=dhi

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, 204204);