#199227

order	512
nilpotency class	3
isoclinism	isoclinic class of #199221
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	#60415	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,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, 106423);