#6454406

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437793
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24402	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1912 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)(3,6)(4,5)(7,8)(9,10,14,21,15,20,16,22)(11,23,26,35,27,36,28,17)(12,29,30,38,31,18,13,19)(24,34,25,33,32,37,39,40);(9,11)(10,17)(12,25)(13,32)(14,28)(15,27)(16,26)(18,37)(19,33)(20,35)(21,36)(22,23)(24,30)(29,34)(31,39)(38,40);(2,5)(6,8)(9,12,15,31)(10,18,20,29)(11,24,27,32)(13,14,30,16)(17,33,35,40)(19,22,38,21)(23,37,36,34)(25,28,39,26);(9,13)(10,19)(11,25)(12,14)(15,30)(16,31)(17,34)(18,22)(20,38)(21,29)(23,33)(24,28)(26,32)(27,39)(35,37)(36,40);(1,3,4,7)(2,6,5,8)(10,20)(17,35)(18,29)(19,38)(21,22)(23,36)(33,40)(34,37)

Polycyclic presentation:

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

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, 6361602);