#6466565

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443052
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
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1862 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)(3,7)(4,5)(6,8)(9,10)(11,20)(12,23)(13,17)(14,19)(15,18)(16,25)(21,26)(22,32)(24,36)(27,34)(28,33)(29,31)(30,38)(35,39)(37,40);(2,5)(6,7)(9,11,20,10,15,19,14,18)(12,16,29,31,25,23,26,21)(13,17,32,34,28,33,27,22)(24,37,40,36,38,35,39,30);(1,3,4,8)(2,6,5,7)(9,12,20,29,15,25,14,26)(10,16,11,21,18,23,19,31)(13,24,32,40,28,38,27,39)(17,30,22,35,33,36,34,37);(9,13,14,27,15,28,20,32)(10,17,19,34,18,33,11,22)(12,24,26,39,25,38,29,40)(16,30,31,37,23,36,21,35);(12,25)(16,23)(21,31)(24,38)(26,29)(30,36)(35,37)(39,40)

Polycyclic presentation:

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

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, 6373761);