#6438615

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437608
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60425	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2126 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,g^c=gi,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, 6345811);