#6452034

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437798
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3010	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2596 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)(3,8)(4,6)(5,7)(9,10,15,20,16,11,14,19)(12,17,30,23,25,21,26,33)(13,18,31,27,29,22,28,34)(24,32,39,36,37,35,38,40);(9,11,16,10)(12,23,25,33)(13,27,29,34)(14,19,15,20)(17,26,21,30)(18,28,22,31)(24,35,37,32)(36,38,40,39);(1,3,5,4)(2,6,7,8)(9,12)(10,17)(11,21)(13,24)(14,26)(15,30)(16,25)(18,32)(19,33)(20,23)(22,35)(27,36)(28,38)(29,37)(31,39)(34,40);(1,4,5,3)(2,6,7,8)(9,13)(10,18)(11,22)(12,24)(14,28)(15,31)(16,29)(17,32)(19,34)(20,27)(21,35)(23,36)(25,37)(26,38)(30,39)(33,40);(12,25)(17,21)(23,33)(24,37)(26,30)(32,35)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,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, 6359230);