#199061

order	512
nilpotency class	3
isoclinism	isoclinic class of #198789
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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1258 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257350, #72257396, #72257455
maximal normal subgroup of	#72257350, #72257396, #72257455
maximal quotient of	#72240832

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106257);