#199236

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	#60559	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1127 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)(3,7)(6,8)(9,10)(11,16,23,31)(12,24)(13,20)(14,18)(15,19)(17,25)(21,35,37,39)(22,33,27,30)(26,34)(28,32)(29,36,40,38);(1,2,3,5,4,6,7,8)(9,11)(10,16,19,31)(12,21)(13,27)(14,22)(15,23)(17,29,32,40)(18,33,20,30)(24,35,34,39)(25,38)(26,37)(28,36);(9,12)(10,17)(11,21)(13,25)(14,28)(15,26)(16,29)(18,24)(19,32)(20,34)(22,36)(23,37)(27,38)(30,39)(31,40)(33,35);(2,6)(5,8)(9,13,15,14)(10,18,19,20)(11,22,23,27)(12,25,26,28)(16,30,31,33)(17,24,32,34)(21,36,37,38)(29,39,40,35)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106432);