#6444255

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2767 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351451);