#199394

order	512
nilpotency class	3
isoclinism	isoclinic class of #199230
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1116 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)(9,10,14,18,15,19,13,28)(11,16,23,30,22,32,21,31)(12,24,26,17,27,33,25,34)(20,35,38,39,37,40,36,29);(1,2,3,5,4,6,7,8)(9,11,14,21,15,22,13,23)(10,16,28,30,19,32,18,31)(12,20,26,36,27,37,25,38)(17,29,24,35,34,39,33,40);(9,12)(10,17)(11,20)(13,25)(14,26)(15,27)(16,29)(18,33)(19,34)(21,36)(22,37)(23,38)(24,28)(30,35)(31,40)(32,39);(2,6)(5,8)(9,13,15,14)(10,18,19,28)(11,21,22,23)(12,25,27,26)(16,30,32,31)(17,33,34,24)(20,36,37,38)(29,35,39,40)

Polycyclic presentation:

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

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, 106590);