#6375903

order	512
nilpotency class	3
isoclinism	isoclinic class of #60768
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60422	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,g^c=gi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283099);