#6433077

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	#60369	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2744 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)(3,5)(4,6)(7,8)(10,15)(11,19)(13,25)(17,27)(18,28)(20,33)(21,22)(24,36)(26,30)(32,38)(34,35)(37,40);(1,3)(2,5)(4,7)(6,8)(9,10,13,15,14,18,25,28)(11,16,21,26,22,31,19,30)(12,17,24,27,23,32,36,38)(20,29,35,37,34,39,33,40);(3,7)(5,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,26)(17,29)(18,31)(19,25)(23,34)(24,35)(27,37)(28,30)(32,39)(33,36)(38,40);(2,6)(5,8)(9,12)(10,17)(11,20)(13,24)(14,23)(15,27)(16,29)(18,32)(19,33)(21,35)(22,34)(25,36)(26,37)(28,38)(30,40)(31,39);(12,23)(17,32)(20,34)(24,36)(27,38)(29,39)(33,35)(37,40)

Polycyclic presentation:

2^9|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,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, 6340273);