#197463

order	512
nilpotency class	3
isoclinism	isoclinic class of #197339
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1263 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104659);