#208747

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×2₋¹⁺⁴	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

981 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=eh,f^b=fh,d^c=dh,e^c=eh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115943);