#6483300

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473174
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1918 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)(3,8)(4,6)(5,7)(9,10,16,13)(11,21,24,36)(12,25,28,35)(14,20,15,19)(17,32,31,23)(18,29,27,30)(22,34,26,33)(37,40,39,38);(3,4)(6,8)(9,11)(10,17)(12,26)(13,31)(14,32)(15,23)(16,24)(18,34)(19,36)(20,21)(22,28)(25,39)(27,33)(29,38)(30,40)(35,37);(1,3,5,4)(2,6,7,8)(9,12,15,30,16,28,14,29)(10,18,20,25,13,27,19,35)(11,22,23,38,24,26,32,40)(17,33,21,37,31,34,36,39);(1,4,5,3)(2,6,7,8)(9,13,16,10)(11,17,24,31)(12,27,28,18)(14,20,15,19)(21,32,36,23)(22,33,26,34)(25,30,35,29)(37,40,39,38);(2,7)(6,8)(10,13)(12,28)(17,31)(19,20)(21,36)(22,26)(29,30)(38,40)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6390496);