#6487502

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470028
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

4075 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)(12,17)(13,20)(14,19)(15,18)(16,24)(22,38)(23,35)(25,37)(26,36)(27,33)(28,31)(29,34)(30,40)(32,39);(3,4)(6,7)(9,11)(10,16)(12,25)(13,24)(14,28)(15,23)(17,32)(18,31)(19,21)(20,35)(22,27)(26,40)(29,39)(30,33)(34,38)(36,37);(1,3,5,4)(2,6,8,7)(9,12,15,27)(10,17,18,33)(11,22,23,25)(13,26,14,29)(16,30,31,32)(19,34,20,36)(21,37,35,38)(24,39,28,40);(1,4,5,3)(2,7,8,6)(10,18)(11,23)(17,33)(19,20)(22,25)(24,28)(34,36)(39,40);(2,8)(6,7)(9,13,15,14)(10,19,18,20)(11,24,23,28)(12,26,27,29)(16,21,31,35)(17,34,33,36)(22,39,25,40)(30,37,32,38)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6394698);