#199471

order	512
nilpotency class	3
isoclinism	isoclinic class of #199212
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

1446 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:

(1,2,4,7,5,8,3,6)(9,10)(11,16,23,32)(12,25)(13,20)(14,18)(15,19)(17,26)(21,35,37,40)(22,31,24,30)(27,34)(28,33)(29,38,39,36);(2,6,8,7)(9,11)(10,16,19,32)(12,21)(13,22)(14,24)(15,23)(17,29,33,39)(18,30,20,31)(25,35,34,40)(26,36)(27,37)(28,38);(9,12)(10,17)(11,21)(13,26)(14,28)(15,27)(16,29)(18,25)(19,33)(20,34)(22,36)(23,37)(24,38)(30,35)(31,40)(32,39);(9,13,15,14)(10,18,19,20)(11,22,23,24)(12,26,27,28)(16,30,32,31)(17,25,33,34)(21,36,37,38)(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=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,g^b=gi,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239929
maximal quotient of	#72239856

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106667);