#304720

order	512
nilpotency class	3
isoclinism	isoclinic class of #271387
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

759 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)(9,10,13,15)(11,19,21,24)(12,18,14,17)(16,22,23,20)(25,26)(27,33)(28,30)(29,37)(31,34)(32,35)(36,39)(38,40);(1,3,5,4)(2,6,8,7)(10,15)(16,23)(17,18)(19,24)(26,30)(27,34)(29,35)(31,39)(33,38)(36,40);(1,3,5,4)(2,7,8,6)(9,11,12,22,13,21,14,20)(10,16,17,24,15,23,18,19)(25,27,35,40,28,36,29,34)(26,31,37,39,30,38,32,33);(1,4,5,3)(2,7,8,6)(9,12,13,14)(10,17,15,18)(11,20,21,22)(16,19,23,24)(26,30)(31,38)(32,37)(33,39)

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,c^2=ef,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,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, 211916);