#6574058

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dh,e^c=ei,e^d=ehi

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, 6481254);