#6450031

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437995
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	#62815	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3090 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:

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

Polycyclic presentation:

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

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, 6357227);