#6433083

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433077
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	#60371	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2712 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)(3,5)(4,6)(7,8)(9,10,15,19)(11,20,22,36)(12,24,28,33)(13,17,29,23)(14,30,31,18)(16,26,34,27)(21,25,37,40)(32,35,38,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,23,15,22,31,17)(10,13,18,20,19,29,30,36)(12,21,26,38,28,37,27,32)(16,25,33,35,34,40,24,39);(3,7)(5,8)(9,12)(10,16)(11,21)(13,25)(14,26)(15,28)(17,32)(18,33)(19,34)(20,35)(22,37)(23,38)(24,30)(27,31)(29,40)(36,39);(2,6)(5,8)(9,13)(10,17)(11,18)(12,25)(14,20)(15,29)(16,32)(19,23)(21,33)(22,30)(24,37)(26,35)(27,39)(28,40)(31,36)(34,38);(11,22)(13,29)(17,23)(20,36)(21,37)(25,40)(32,38)(35,39)

Polycyclic presentation:

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

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, 6340279);