#295060

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

779 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)(3,7)(4,5)(6,8)(9,10,12,15)(11,17,19,24)(13,21,18,16)(14,22,23,20)(25,26,29,31)(27,35,38,40)(28,34,30,33)(32,36,39,37);(2,5)(6,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,23)(22,24)(26,31)(27,36)(28,30)(32,35)(37,38)(39,40);(9,11,18,22,12,19,13,20)(10,14,21,24,15,23,16,17)(25,27,30,36,29,38,28,37)(26,32,34,40,31,39,33,35);(1,3,4,8)(2,6,5,7)(10,15)(11,19)(16,21)(20,22)(25,28,29,30)(26,33,31,34)(27,37,38,36)(32,35,39,40)

Polycyclic presentation:

2^9|a^2=h,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, 202256);