#6446744

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437995
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	#62812	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=gi,e^2=i,b^a=bf,c^a=cg,e^a=ei,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6353940);