#197620

order	512
nilpotency class	3
isoclinism	isoclinic class of #19420
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1119 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,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, 104816);