#199233

order	512
nilpotency class	3
isoclinism	isoclinic class of #199230
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

1252 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)(3,4)(5,8)(6,7)(9,10)(11,20,23,32)(12,24)(13,19)(14,28)(15,18)(16,22,31,27)(17,25)(21,36,39,30)(26,34)(29,33)(35,37,40,38);(1,3,4,2,6,8,5,7)(9,11,13,27,15,23,14,22)(10,16,19,32,18,31,28,20)(12,21,25,38,26,39,29,37)(17,30,34,40,33,36,24,35);(9,12)(10,17)(11,21)(13,25)(14,29)(15,26)(16,30)(18,33)(19,34)(20,35)(22,37)(23,39)(24,28)(27,38)(31,36)(32,40);(1,4,6,5)(2,3,7,8)(10,18)(16,31)(17,33)(19,28)(20,32)(24,34)(30,36)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=hi,f^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.

Relationships

maximal subgroup of	#72257594, #72316979
maximal normal subgroup of	#72257594, #72316979
maximal quotient of	#72257678

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106429);