#6490950

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

3180 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)(12,17)(13,20)(14,18)(15,19)(16,24)(22,38)(23,32)(25,31)(26,37)(27,36)(28,35)(29,34)(30,39)(33,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,26)(13,24)(14,23)(15,25)(17,33)(18,21)(19,32)(20,31)(22,28)(27,39)(29,40)(30,35)(34,38)(36,37);(2,5)(6,8)(9,12,15,28)(10,17,19,35)(11,22,25,26)(13,27,14,29)(16,30,32,33)(18,34,20,36)(21,37,31,38)(23,39,24,40);(2,5)(6,8)(9,13,15,14)(10,18,19,20)(11,23,25,24)(12,27,28,29)(16,31,32,21)(17,34,35,36)(22,39,26,40)(30,38,33,37);(1,3,4,7)(2,6,5,8)(10,19)(16,32)(17,35)(18,20)(21,31)(30,33)(34,36)(37,38)

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=hi,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, 6398146);