#6487527

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

4081 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=i,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, 6394723);