#6685526

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682638
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	#60771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1795 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=g,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592722);