#6576753

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
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	#60377	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2578 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)(11,20)(12,17)(13,18)(14,16)(15,23)(19,30)(21,32)(22,37)(24,33)(25,40)(26,36)(27,29)(28,34)(31,39)(35,38);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,25)(13,23)(14,21)(16,33)(17,35)(18,20)(19,32)(22,29)(24,30)(26,38)(27,31)(28,39)(34,37)(36,40);(10,16)(11,21)(12,26)(14,30)(15,20)(17,27)(18,19)(22,38)(23,24)(25,39)(34,36)(35,40);(1,4,5,3)(2,6,7,8)(9,10)(11,15)(12,27)(13,18)(14,19)(16,30)(17,29)(20,23)(21,32)(22,35)(24,33)(25,31)(26,34)(28,36)(37,38)(39,40);(9,12,13,26)(10,17,18,36)(11,22,23,39)(14,28,30,29)(15,31,20,37)(16,34,19,27)(21,25,24,38)(32,40,33,35)

Polycyclic presentation:

2^9|b^2=fi,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6483949);