#6465491

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2636 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6372687);