#6483303

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
prime radical	#60808	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1902 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)(10,16)(11,20)(13,14)(17,31)(18,30)(19,28)(21,35)(22,23)(24,36)(26,27)(32,33)(34,38);(3,4)(6,8)(9,10)(11,17)(12,18)(13,28)(14,16)(15,19)(20,29)(21,32)(22,31)(23,37)(24,33)(25,34)(26,38)(27,30)(35,39)(36,40);(1,3,5,4)(2,6,7,8)(9,11,14,20,15,22,13,23)(10,17,16,29,19,31,28,37)(12,24,27,36,25,21,26,35)(18,33,30,40,34,32,38,39);(1,4,5,3)(2,6,7,8)(9,12,15,25)(10,18,19,34)(11,21,22,24)(13,26,14,27)(16,30,28,38)(17,32,31,33)(20,35,23,36)(29,39,37,40);(2,7)(6,8)(11,22)(12,25)(17,31)(18,34)(20,23)(26,27)(29,37)(30,38)

Polycyclic presentation:

2^9|c^2=g,d^2=hi,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,d^c=di,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, 6390499);