#390151

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

937 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,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, 297347);