#6490972

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470058
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi

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, 6398168);