#6438607

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2098 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)(11,21)(12,27)(13,20)(14,19)(15,18)(16,25)(17,29)(22,34)(23,26)(24,33)(28,37)(30,36)(31,35)(32,39)(38,40);(2,5)(6,8)(9,11)(10,16)(12,28)(13,24)(14,25)(15,26)(17,34)(18,33)(19,21)(20,23)(22,31)(27,32)(29,38)(30,39)(35,40)(36,37);(1,3,4,7)(2,6,5,8)(9,12,14,30)(10,17,19,35)(11,22,25,38)(13,29,15,31)(16,32,21,37)(18,27,20,36)(23,34,33,40)(24,28,26,39);(9,10)(11,23)(12,17)(13,18)(14,19)(15,20)(16,24)(21,26)(22,34)(25,33)(27,29)(28,32)(30,35)(31,36)(37,39)(38,40);(9,13,14,15)(10,18,19,20)(11,24,25,26)(12,29,30,31)(16,33,21,23)(17,27,35,36)(22,28,38,39)(32,40,37,34)

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,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, 6345803);